Adaptive basis selection for encoded fusion measurements

ABSTRACT

A quantum computing system and methods for performing fusion based quantum computing on encoded qubits. A fusion controller sequentially performs a series of fusion measurements on respective physical qubits of first and second encoded qubits to obtain a respective series of classical measurement results. For respective fusion measurements of the series of fusion measurements, a basis for performing the respective fusion measurement is selected based on classical measurement results of previous fusion measurements. An encoded fusion measurement result is determined based on the classical measurement results, and the encoded fusion measurement result is stored in a memory medium.

PRIORITY INFORMATION

This application claims priority to U.S. provisional patent applicationSer. No. 63/001,745, entitled “Adaptive Fusion,” and filed Mar. 30,2020; U.S. provisional patent application Ser. No. 63/009,920, entitled“Fusion Based Quantum Computing with Kagome Lattice,” and filed Apr. 14,2020; U.S. provisional patent application Ser. No. 63/081,691, entitled“Adaptive Basis Selection for Fusion Measurements,” and filed Sep. 22,2020; U.S. provisional patent application Ser. No. 63/118,319, entitled“Encoded Fusion Measurements with Local Adaptivity,” and filed Nov. 25,2020; and U.S. provisional patent application Ser. No. 63/119,395,entitled “Adaptive Basis Selection for Fusion Measurements,” and filedNov. 30, 2020, which are all hereby incorporated by reference in theirentirety as though fully and completely set forth herein.

TECHNICAL FIELD

Embodiments herein relate generally to quantum computational devices,such as photonic devices (or hybrid electronic/photonic devices) forperforming encoded fusion measurements to generate fault tolerantquantum computing devices and associated methods.

BACKGROUND

Quantum computing can be distinguished from “classical” computing by itsreliance on structures referred to as “qubits.” At the most generallevel, a qubit is a quantum system that may exist in one of twoorthogonal states (denoted as |0

and |1

in the conventional bra/ket notation) or in a superposition of the twostates

${e.g.},{\frac{1}{\sqrt{2}}{\left( {\left. 0 \right\rangle + \left. 1 \right\rangle} \right).}}$By operating on a system (or ensemble) of qubits, a quantum computer mayquickly perform certain categories of computations that would requireimpractical amounts of time in a classical computer.

In fault tolerant quantum computing, quantum error correction isrequired to avoid an accumulation of qubit errors that then leads toerroneous computational outcomes. One method of achieving faulttolerance is to employ error correcting codes (e.g., topological codes)for quantum error correction. More specifically, a collection ofphysical qubits may be generated in an entangled state (also referred toherein as an error correcting code) that encodes for a single logicalqubit that is protected from errors.

In some quantum computing systems, cluster states of multiple qubits,or, more generally, graph states may be used as the error correctingcode. A graph state is a highly entangled multi-qubit state that may berepresented visually as a graph with nodes representing qubits and edgesrepresenting entanglement between the qubits.

One of the main barriers to widespread use of quantum technologies, suchas quantum computing, quantum communications, and the like, is theability to reliably generate entanglement among two or more physicalquantum systems, e.g., between two or more qubits. Various problems thateither inhibit the generation of entangled states or destroy theentanglement once created (e.g., such as decoherence) have frustratedadvancements in quantum technologies that rely on the use of highlyentangled quantum states. Furthermore, in some qubit architectures,e.g., photonic architectures, the generation of entangled states ofmultiple qubits is an inherently probabilistic process that may have alow probability of success. For example, current methods for producingBell states from single photons have success probabilities of around 20%(corresponding to an 80% failure rate). Accordingly, there remains aneed for improved systems and methods for producing entangled states andquantum error correcting codes.

SUMMARY

Some embodiments described herein include quantum computing devices,systems and methods for performing fusion based quantum computing onencoded qubits.

In some embodiments, a photonic quantum computing system includes anon-transitory computer-readable memory medium, a plurality of encodedqubits each comprising a plurality of physical qubits, a fusioncontroller, and a plurality of fusion sites coupled to the fusioncontroller.

In some embodiments, the photonic quantum computing system is configuredto sequentially performs a series of fusion measurements on respectivephysical qubits of first and second encoded qubits to obtain arespective series of classical measurement results.

For respective fusion measurements of the series of fusion measurements,a basis for performing the respective fusion measurement is selectedbased on classical measurement results of previous fusion measurements.

An encoded fusion measurement result is determined based on theclassical measurement results, and the encoded fusion measurement resultis stored in a memory medium.

The techniques described herein may be implemented in and/or used with anumber of different types of devices, including but not limited tophotonic quantum computing devices and/or systems, hybridquantum/classical computing systems, and any of various other quantumcomputing systems.

This Summary is intended to provide a brief overview of some of thesubject matter described in this document. Accordingly, it will beappreciated that the above-described features are merely examples andshould not be construed to narrow the scope or spirit of the subjectmatter described herein in any way. Other features, aspects, andadvantages of the subject matter described herein will become apparentfrom the following Detailed Description, Figures, and Claims.

BRIEF DESCRIPTION OF THE DRAWINGS

For a better understanding of the various described embodiments,reference should be made to the Detailed Description below, inconjunction with the following drawings in which like reference numeralsrefer to corresponding parts throughout the Figures.

FIG. 1 shows two representations of a portion of a pair of waveguidescorresponding to a dual-rail-encoded photonic qubit;

FIG. 2A shows a schematic diagram for coupling of two modes;

FIG. 2B shows, in schematic form, a physical implementation of modecoupling in a photonic system that may be used in some embodiments;

FIGS. 3A and 3B show, in schematic form, examples of physicalimplementations of a Mach-Zehnder Interferometer (MZI) configurationthat may be used in some embodiments;

FIG. 4A shows another schematic diagram for coupling of two modes;

FIG. 4B shows, in schematic form, a physical implementation of the modecoupling of FIG. 4A in a photonic system that may be used in someembodiments;

FIG. 5 shows a four-mode coupling scheme that implements a “spreader,”or “mode-information erasure,” transformation on four modes inaccordance with some embodiments;

FIG. 6 illustrates an example optical device that may implement thefour-mode mode-spreading transform shown schematically in FIG. 5 inaccordance with some embodiments;

FIG. 7 shows a circuit diagram for a dual-rail-encoded Bell stategenerator that may be used in some embodiments;

FIG. 8A shows a circuit diagram for a dual-rail-encoded type I fusiongate that may be used in some embodiments;

FIG. 8B shows example results of type I fusion operations using the gateof FIG. 8A;

FIG. 9A shows a circuit diagram for a dual-rail-encoded type II fusiongate that may be used in some embodiments;

FIG. 9B shows an example result of a type II fusion operation using thegate of FIG. 9A;

FIG. 10 illustrates an example of a qubit entangling system 1001 inaccordance with some embodiments;

FIGS. 11A-11C are diagrams illustrating a cluster state and acorresponding syndrome graph for an entangled state of physical qubitsin accordance with some embodiments;

FIG. 12 shows a quantum computing system in accordance with one or moreembodiments;

FIG. 13 is a schematic illustration of quantum computing systemincluding a qubit fusion system in accordance with some embodiments;

FIG. 14 shows one example of qubit fusion system in accordance with someembodiments;

FIG. 15 shows one possible example of a fusion site as configured tooperate with a fusion controller to provide measurement outcomes to adecoder for fault tolerant quantum computation in accordance with someembodiments;

FIGS. 16A-B illustrate resource states and unit cells for fault tolerantquantum computation in accordance with one or more embodiments;

FIGS. 16C-E illustrate the primal and dual syndrome graphs for anexample logical qubit in accordance with one or more embodiments;

FIGS. 17A-D illustrate connected components of primal and dual syndromegraphs in two dimensions in accordance with some embodiments;

FIG. 18 illustrates erasures in a three-dimensional syndrome graph inaccordance with some embodiments;

FIG. 19 is a schematic illustration of an edge that connects twoportions of a single connected component, according to some embodiments;

FIG. 20 is a schematic illustration of a growth region of a connectedcomponent, according to some embodiments;

FIG. 21 is a graph illustrating photon loss thresholds while performingand not performing adaptive basis selection during a fusion measurementsequence, according to some embodiments;

FIG. 22 is a graph illustrating photon loss thresholds while performingadaptive basis selection and always boosting, performing adaptive basisselection and adaptive boosting, and not performing adaptive basisselection during a fusion measurement sequence, according to someembodiments;

FIG. 23 is a schematic illustration of a typical fusion measurement,according to some embodiments;

FIG. 24A illustrates a fusion measurement where the two qubits to befused are encoded qubits;

FIG. 24B illustrates an example of how the encoded qubits of FIG. 24Amay be entangled, according to some embodiments;

FIG. 25 illustrates a system setup for performing a fusion measurementin one of four different manners, according to some embodiments;

FIG. 26 is a flow chart diagram illustrating a method for performingadaptive basis selection during an encoded fusion measurement, accordingto some embodiments;

FIG. 27 is a schematic illustration of a qubit fusion system that may beused to perform method steps of some embodiments;

FIG. 28 illustrates a three-way encoded fusion measurement, according tosome embodiments;

FIG. 29 is a flow chart diagram illustrating a method for performingadaptive basis selection during a three-way encoded fusion measurement,according to some embodiments;

FIG. 30 shows an example of a Type II fusion circuit for a polarizationencoding, according to some embodiments;

FIG. 31 shows an example of a Type II fusion circuit for a pathencoding, according to some embodiments;

FIGS. 32A-D shows effects of fusion in the generation of a clusterstate, according to some embodiments;

FIG. 33 shows examples of Type II fusion gates boosted once inpolarization and path encodings, according to some embodiments;

FIG. 34 shows a table with the effects of a few rotated variations ofthe Type II fusion gate used to fuse two small entangled states,according to some embodiments;

FIG. 35 shows examples of Type II fusion gate implementations for a pathencoding, according to some embodiments;

FIG. 36A is a diagram illustrating a 6-qubit ring resource state,according to some embodiments; and

FIG. 36B is a diagram illustrated a 24 qubit encoded qubit resourcestate corresponding to the 6-qubit ring resource state of FIG. 36A.

While the features described herein may be susceptible to variousmodifications and alternative forms, specific embodiments thereof areshown by way of example in the drawings and are herein described indetail. It should be understood, however, that the drawings and detaileddescription thereto are not intended to be limiting to the particularform disclosed, but on the contrary, the intention is to cover allmodifications, equivalents and alternatives falling within the spiritand scope of the subject matter as defined by the appended claims.

DETAILED DESCRIPTION

Disclosed herein are examples (also referred to as “embodiments”) ofsystems and methods for creating qubits and superposition states(including entangled states) of qubits based on various physical quantumsystems, including photonic systems. Such embodiments may be used, forexample, in quantum computing as well as in other contexts (e.g.,quantum communication) that exploit quantum entanglement. To facilitateunderstanding of the disclosure, an overview of relevant concepts andterminology is provided in Section 1. With this context established,Section 2 describes systems and methods for performing adaptive basisselection while performing fusion measurements in a quantum errorcorrecting code.

Although embodiments are described with specific detail to facilitateunderstanding, those skilled in the art with access to this disclosurewill appreciate that the claimed invention may be practiced withoutthese details. Reference will now be made in detail to embodiments,examples of which are illustrated in the accompanying drawings. In otherinstances, well-known methods, procedures, components, circuits, andnetworks have not been described in detail so as not to unnecessarilyobscure aspects of the embodiments.

As used herein, the term “syndrome” and/or “syndrome state” refers to aset of classical information (e.g. data represented by digital valuessuch as ones and zeros) that results when a series of measurements (e.g.stabilizer measurements) are applied to the physical qubits of thecluster state that makes up the error correcting code. As described infurther detail below, these measurement outcomes may be represented byclassical data and, based on the knowledge of the particular geometry ofthe cluster state/error correcting code, may be used to determine aclassical data structure referred to herein as a “syndrome graph.”

Section I. Overview of Quantum Computing

Quantum computing relies on the dynamics of quantum objects, e.g.,photons, electrons, atoms, ions, molecules, nanostructures, and thelike, which follow the rules of quantum theory. In quantum theory, thequantum state of a quantum object is described by a set of physicalproperties, the complete set of which is referred to as a mode. In someembodiments, a mode is defined by specifying the value (or distributionof values) of one or more properties of the quantum object. For example,in the case where the quantum object is a photon, modes may be definedby the frequency of the photon, the position in space of the photon(e.g., which waveguide or superposition of waveguides the photon ispropagating within), the associated direction of propagation (e.g., thek-vector for a photon in free space), the polarization state of thephoton (e.g., the direction (horizontal or vertical) of the photon'selectric and/or magnetic fields), a time window in which the photon ispropagating, the orbital angular momentum state of the photon, and thelike.

For the case of photons propagating in a waveguide, it is convenient toexpress the state of the photon as one of a set of discretespatio-temporal modes. For example, the spatial mode k_(i) of the photonis determined according to which one of a finite set of discretewaveguides the photon is propagating in, and the temporal mode t_(j) isdetermined by which one of a set of discrete time periods (referred toherein as “bins”) the photon is present in. In some photonicimplementations, the degree of temporal discretization may be providedby a pulsed laser which is responsible for generating the photons. Inexamples below, spatial modes will be used primarily to avoidcomplication of the description. However, one of ordinary skill willappreciate that the systems and methods may apply to any type of mode,e.g., temporal modes, polarization modes, and any other mode or set ofmodes that serves to specify the quantum state. Further, in thedescription that follows, embodiments will be described that employphotonic waveguides to define the spatial modes of the photon. However,persons of ordinary skill in the art with access to this disclosure willappreciate that other types of mode, e.g., temporal modes, energystates, and the like, may be used without departing from the scope ofthe present disclosure. In addition, persons of ordinary skill in theart will be able to implement examples using other types of quantumsystems, including but not limited to other types of photonic systems.

For quantum systems of multiple indistinguishable particles, rather thandescribing the quantum state of each particle in the system, it isuseful to describe the quantum state of the entire many-body systemusing the formalism of Fock states (sometimes referred to as theoccupation number representation). In the Fock state description, themany-body quantum state is specified by how many particles there are ineach mode of the system. For example, a multi-mode, two particle Fockstate |1001

_(1,2,3,4) specifies a two-particle quantum state with one particle inmode 1, zero particles in mode 2, zero particles in mode 3, and oneparticle in mode 4. Again, as introduced above, a mode may be anyproperty of the quantum object. For the case of a photon, any two modesof the electromagnetic field may be used, e.g., one may design thesystem to use modes that are related to a degree of freedom that may bemanipulated passively with linear optics. For example, polarization,spatial degree of freedom, or angular momentum could be used. Thefour-mode system represented by the two particle Fock state |1001

_(1,2,3,4) may be physically implemented as four distinct waveguideswith two of the four waveguides having one photon travelling withinthem. Other examples of a state of such a many-body quantum systeminclude the four-particle Fock state |1111

_(1,2,3,4) that represents each mode occupied by one particle and thefour-particle Fock state |2200

_(1,2,3,4) that represents modes 1 and 2 respectively occupied by twoparticles and modes 3 and 4 occupied by zero particles. For modes havingzero particles present, the term “vacuum mode” is used. For example, forthe four-particle Fock state |2200

_(1,2,3,4) modes 3 and 4 are referred to herein as “vacuum modes.” Fockstates having a single occupied mode may be represented in shorthandusing a subscript to identify the occupied mode. For example, |0010

_(1,2,3,4) is equivalent to |1₃

.

1.1 Qubits

As used herein, a “qubit” (or quantum bit) is a quantum system with anassociated quantum state that may be used to encode information. Aquantum state may be used to encode one bit of information if thequantum state space can be modeled as a (complex) two-dimensional vectorspace, with one dimension in the vector space being mapped to logicalvalue 0 and the other to logical value 1. In contrast to classical bits,a qubit may have a state that is a superposition of logical values 0and 1. More generally, a “qudit” describes any quantum system having aquantum state space that may be modeled as a (complex) n-dimensionalvector space (for any integer n), which may be used to encode n bits ofinformation. For the sake of clarity of description, the term “qubit” isused herein, although in some embodiments the system may also employquantum information carriers that encode information in a manner that isnot necessarily associated with a binary bit, such as a qudit.

Qubits (or qudits) may be implemented in a variety of quantum systems.Examples of qubits include: polarization states of photons; presence ofphotons in waveguides; or energy states of molecules, atoms, ions,nuclei, or photons. Other examples include other engineered quantumsystems such as flux qubits, phase qubits, or charge qubits (e.g.,formed from a superconducting Josephson junction); topological qubits(e.g., Majorana fermions); or spin qubits formed from vacancy centers(e.g., nitrogen vacancies in diamond).

A qubit may be “dual-rail encoded” such that the logical value of thequbit is encoded by occupation of one of two modes of the quantumsystem. For example, the logical 0 and 1 values may be encoded asfollows:|0

_(L)=|10

_(1,2)  (1)|1

_(L)=|01

_(1,2)  (2)where the subscript “L” indicates that the ket represents a logicalstate (e.g., a qubit value) and, as before, the notation |ij

_(1,2) on the right-hand side of the equations above indicates thatthere are i particles in a first mode and j particles in a second mode,respectively (e.g., where i and j are integers). In this notation, atwo-qubit system having a logical state |0

|1

_(L) (representing a state of two qubits, the first qubit being in a ‘0’logical state and the second qubit being in a ‘1’ logical state) may berepresented using occupancy across four modes by |1001

_(1,2,3,4) (e.g., in a photonic system, one photon in a first waveguide,zero photons in a second waveguide, zero photons in a third waveguide,and one photon in a fourth waveguide). In some instances throughout thisdisclosure, the various subscripts are omitted to avoid unnecessarymathematical clutter.1.2 Entangled States

Many of the advantages of quantum computing relative to “classical”computing (e.g., conventional digital computers using binary logic) stemfrom the ability to create entangled states of multi-qubit systems. Inmathematical terms, a state |ψ

of n quantum objects is a separable state if |ψ

=|ψ₁

⊗ . . . ⊗|ψ_(n)

, and an entangled state is a state that is not separable. One exampleis a Bell state, which, loosely speaking, is a type of maximallyentangled state for a two-qubit system, and qubits in a Bell state maybe referred to as a Bell pair. For example, for qubits encoded by singlephotons in pairs of modes (a dual-rail encoding), examples of Bellstates include:

$\begin{matrix}{\left. \Phi^{+} \right\rangle = {\frac{{\left. 0 \right\rangle_{L}\left. 0 \right\rangle_{L}} + {\left. 1 \right\rangle_{L}\left. 1 \right\rangle_{L}}}{\sqrt{2}} = \frac{{\left. 10 \right\rangle\left. 10 \right\rangle} + {\left. 01 \right\rangle\left. 01 \right\rangle}}{\sqrt{2}}}} & (3) \\{\left. \Phi^{-} \right\rangle = {\frac{{\left. 0 \right\rangle_{L}\left. 0 \right\rangle_{L}} - {\left. 1 \right\rangle_{L}\left. 1 \right\rangle_{L}}}{\sqrt{2}} = \frac{{\left. 10 \right\rangle\left. 10 \right\rangle} - {\left. 01 \right\rangle\left. 01 \right\rangle}}{\sqrt{2}}}} & (4) \\{\left. \Psi^{+} \right\rangle = {\frac{{\left. 0 \right\rangle_{L}\left. 1 \right\rangle_{L}} + {\left. 1 \right\rangle_{L}\left. 0 \right\rangle_{L}}}{\sqrt{2}} = \frac{{\left. 10 \right\rangle\left. 01 \right\rangle} + {\left. 01 \right\rangle\left. 10 \right\rangle}}{\sqrt{2}}}} & (5) \\{\left. \Psi^{-} \right\rangle = {\frac{{\left. 0 \right\rangle_{L}\left. 1 \right\rangle_{L}} - {\left. 1 \right\rangle_{L}\left. 0 \right\rangle_{L}}}{\sqrt{2}} = \frac{{\left. 10 \right\rangle\left. 01 \right\rangle} - {\left. 01 \right\rangle\left. 10 \right\rangle}}{\sqrt{2}}}} & (6)\end{matrix}$

More generally, an n-qubit Greenberger-Horne-Zeilinger (GHZ) state (or“n-GHZ state”) is an entangled quantum state of n qubits. For a givenorthonormal logical basis, an n-GHZ state is a quantum superposition ofall qubits being in a first basis state superposed with all qubits beingin a second basis state:

$\begin{matrix}{\left. {GHZ} \right\rangle = \frac{\left. 0 \right\rangle^{\otimes M} + \left. 1 \right\rangle^{\otimes M}}{\sqrt{2}}} & (7)\end{matrix}$where the kets above refer to the logical basis. For example, for qubitsencoded by single photons in pairs of modes (a dual-rail encoding), a3-GHZ state may be written:

$\begin{matrix}{\left. {GHZ} \right\rangle = {\frac{{\left. 0 \right\rangle_{L}\left. 0 \right\rangle_{L}\left. 0 \right\rangle_{L}} - {\left. 1 \right\rangle_{L}\left. 1 \right\rangle_{L}\left. 1 \right\rangle_{L}}}{\sqrt{2}} = \frac{{\left. 10 \right\rangle\left. 10 \right\rangle\left. 10 \right\rangle} + {\left. 01 \right\rangle\left. 01 \right\rangle\left. 01 \right\rangle}}{\sqrt{2}}}} & (8)\end{matrix}$where the kets above refer to photon occupation number in six respectivemodes (with mode subscripts omitted).1.3 Physical Implementations

Qubits (and operations on qubits) may be implemented using a variety ofphysical systems. In some examples described herein, qubits are providedin an integrated photonic system employing waveguides, beam splitters,photonic switches, and single photon detectors, and the modes that maybe occupied by photons are spatiotemporal modes that correspond topresence of a photon in a waveguide. Modes may be coupled using modecouplers, e.g., optical beam splitters, to implement transformationoperations, and measurement operations may be implemented by couplingsingle-photon detectors to specific waveguides. One of ordinary skill inthe art with access to this disclosure will appreciate that modesdefined by any appropriate set of degrees of freedom, e.g., polarizationmodes, temporal modes, and the like, may be used without departing fromthe scope of the present disclosure. For instance, for modes that onlydiffer in polarization (e.g., horizontal (H) and vertical (V)), a modecoupler may be any optical element that coherently rotates polarization,e.g., a birefringent material such as a waveplate. For other systemssuch as ion trap systems or neutral atom systems, a mode coupler may beany physical mechanism that couples two modes, e.g., a pulsedelectromagnetic field that is tuned to couple two internal states of theatom/ion.

In some embodiments of a photonic quantum computing system usingdual-rail encoding, a qubit may be implemented using a pair ofwaveguides. FIG. 1 shows two representations (100, 100′) of a portion ofa pair of waveguides 102, 104 that may be used to provide adual-rail-encoded photonic qubit. At 100, a photon 106 is in waveguide102 and no photon is in waveguide 104 (also referred to as a vacuummode); in some embodiments, this corresponds to the |0

_(L) state of a photonic qubit. At 100′, a photon 108 is in waveguide104, and no photon is in waveguide 102; in some embodiments thiscorresponds to the |1

_(L) state of the photonic qubit. To prepare a photonic qubit in a knownlogical state, a photon source (not shown) may be coupled to one end ofone of the waveguides. The photon source may be operated to emit asingle photon into the waveguide to which it is coupled, therebypreparing a photonic qubit in a known state. Photons travel through thewaveguides, and by periodically operating the photon source, a quantumsystem having qubits whose logical states map to different temporalmodes of the photonic system may be created in the same pair ofwaveguides. In addition, by providing multiple pairs of waveguides, aquantum system having qubits whose logical states correspond todifferent spatiotemporal modes may be created. It should be understoodthat the waveguides in such a system need not have any particularspatial relationship to each other. For instance, they may be but neednot be arranged in parallel.

Occupied modes may be created by using a photon source to generate aphoton that then propagates in the desired waveguide. A photon sourcemay be, for instance, a resonator-based source that emits photon pairs,also referred to as a heralded single photon source. In one example ofsuch a source, the source is driven by a pump, e.g., a light pulse, thatis coupled into a system of optical resonators that, through a nonlinearoptical process (e.g., spontaneous four wave mixing (SFWM), spontaneousparametric down-conversion (SPDC), second harmonic generation, or thelike), may generate a pair of photons. Many different types of photonsources may be employed. Examples of photon pair sources may include amicro-ring-based spontaneous four wave mixing (SPFW) heralded photonsource (HPS). However, the precise type of photon source used is notcritical and any type of nonlinear source, employing any process, suchas SPFW, SPDC, or any other process may be used. Other classes ofsources that do not necessarily require a nonlinear material may also beemployed, such as those that employ atomic and/or artificial atomicsystems, e.g., quantum dot sources, color centers in crystals, and thelike. In some cases, sources may or may not be coupled to photoniccavities, e.g., as may be the case for artificial atomic systems such asquantum dots coupled to cavities. Other types of photon sources alsoexist for SPWM and SPDC, such as optomechanical systems and the like.

In such cases, operation of the photon source may be non-deterministic(also sometimes referred to as “stochastic”) such that a given pumppulse may or may not produce a photon pair. In some embodiments,coherent spatial and/or temporal multiplexing of severalnon-deterministic sources (referred to herein as “active” multiplexing)may be used to allow the probability of having one mode become occupiedduring a given cycle to approach unity. One of ordinary skill willappreciate that many different active multiplexing architectures thatincorporate spatial and/or temporal multiplexing are possible. Forinstance, active multiplexing schemes that employ log-tree, generalizedMach-Zehnder interferometers, multimode interferometers, chainedsources, chained sources with dump-the-pump schemes, asymmetricmulti-crystal single photon sources, or any other type of activemultiplexing architecture may be used. In some embodiments, the photonsource may employ an active multiplexing scheme with quantum feedbackcontrol and the like. In some embodiments described below, use ofmulti-rail encoding allows the probability of a band having one mode tobecome occupied during a given pulse cycle to approach unity withoutactive multiplexing.

Measurement operations may be implemented by coupling a waveguide to asingle-photon detector that generates a classical signal (e.g., adigital logic signal) indicating that a photon has been detected by thedetector. Any type of photodetector that has sensitivity to singlephotons may be used. In some embodiments, detection of a photon (e.g.,at the output end of a waveguide) indicates an occupied mode whileabsence of a detected photon may indicate an unoccupied mode.

Some embodiments described below relate to physical implementations ofunitary transform operations that couple modes of a quantum system,which may be understood as transforming the quantum state of the system.For instance, if the initial state of the quantum system (prior to modecoupling) is one in which one mode is occupied with probability 1 andanother mode is unoccupied with probability 1 (e.g., a state |10

in the Fock notation introduced above), mode coupling may result in astate in which both modes have a nonzero probability of being occupied,e.g., a state a₁|10

+a₂|01

, where |a₁|²+|a₂|²=1. In some embodiments, operations of this kind maybe implemented by using beam splitters to couple modes together andvariable phase shifters to apply phase shifts to one or more modes. Theamplitudes a₁ and a₂ depend on the reflectivity (or transmissivity) ofthe beam splitters and on any phase shifts that are introduced.

FIG. 2A shows a schematic diagram 210 (also referred to as a circuitdiagram or circuit notation) for coupling of two modes. The modes aredrawn as horizontal lines 212, 214, and the mode coupler 216 isindicated by a vertical line that is terminated with nodes (solid dots)to identify the modes being coupled. In the more specific language oflinear quantum optics, the mode coupler 216 shown in FIG. 2A representsa 50/50 beam splitter that implements a transfer matrix:

$\begin{matrix}{{T = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 & i \\i & 1\end{pmatrix}}},} & (9)\end{matrix}$where T defines the linear map for the photon creation operators on twomodes. (In certain contexts, transfer matrix T may be understood asimplementing a first-order imaginary Hadamard transform.) By conventionthe first column of the transfer matrix corresponds to creationoperators on the top mode (referred to herein as mode 1, labeled ashorizontal line 212), and the second column corresponds to creationoperators on the second mode (referred to herein as mode 2, labeled ashorizontal line 214), and so on if the system includes more than twomodes. More explicitly, the mapping may be written as:

$\begin{matrix}{\left. \begin{pmatrix}a_{1}^{\dagger} \\a_{2}^{\dagger}\end{pmatrix}_{input}\mapsto{\frac{1}{\sqrt{2}}\begin{pmatrix}1 & {- i} \\{- i} & 1\end{pmatrix}\begin{pmatrix}a_{1}^{\dagger} \\a_{2}^{\dagger}\end{pmatrix}_{output}} \right.,} & (10)\end{matrix}$where subscripts on the creation operators indicate the mode that isoperated on, the subscripts input and output identify the form of thecreation operators before and after the beam splitter, respectively andwhere:a _(i) |n _(i) ,n _(j)

=√{square root over (n _(i))}|n _(i)−1,n _(j)

a _(j) |n _(i) ,n _(j)

=√{square root over (n _(j))}|n _(i) ,n _(j)−1

a _(i) ^(†) |n _(i) ,n _(j)

=√{square root over (n _(i)+1)}|n _(i)+1,n _(j)

a _(j) ^(†) |n _(i) ,n _(j)

=√{square root over (n _(j)+1)}|n _(i) ,n _(j)+1

  (11)For example, the application of the mode coupler shown in FIG. 2A leadsto the following mappings:

$\begin{matrix}{\left. a_{1_{input}}^{\dagger}\mapsto{\frac{1}{\sqrt{2}}\left( {a_{1_{output}}^{\dagger} - {ia_{2_{output}}^{\dagger}}} \right)} \right.\left. a_{2_{input}}^{\dagger}\mapsto{\frac{1}{\sqrt{2}}\left( {{{- i}a_{1_{output}}^{\dagger}} + a_{2_{output}}^{\dagger}} \right)} \right.} & (12)\end{matrix}$Thus, the action of the mode coupler described by Eq. (9) is to take theinput states |10

, |01

, and |11

to

$\begin{matrix}{\left. \left. 10 \right\rangle\mapsto\frac{\left. 10 \right\rangle - {i\left. 01 \right\rangle}}{\sqrt{2}} \right.\left. \left. 01 \right\rangle\mapsto\frac{{{- i}\left. 10 \right\rangle} + \left. 01 \right\rangle}{\sqrt{2}} \right.\left. \left. 11 \right\rangle\mapsto{{{- i}/2}\left( {\left. 20 \right\rangle + \left. 02 \right\rangle} \right)} \right.} & (13)\end{matrix}$

FIG. 2B shows a physical implementation of a mode coupling thatimplements the transfer matrix T of Eq. (9) for two photonic modes inaccordance with some embodiments. In this example, the mode coupling isimplemented using a waveguide beam splitter 200, also sometimes referredto as a directional coupler or mode coupler. A waveguide beam splitter200 may be realized by bringing two waveguides 202, 204 into closeenough proximity that the evanescent field of one waveguide may coupleinto the other. By adjusting the separation d between waveguides 202,204 and/or the length l of the coupling region, different couplingsbetween modes may be obtained. In this manner, a waveguide beam splitter200 may be configured to have a desired transmissivity. For example, thebeam splitter may be engineered to have a transmissivity equal to 0.5(i.e., a 50/50 beam splitter for implementing the specific form of thetransfer matrix T introduced above). If other transfer matrices aredesired, the reflectivity (or the transmissivity) may be engineered tobe greater than 0.6, greater than 0.7, greater than 0.8, or greater than0.9 without departing from the scope of the present disclosure.

In addition to mode coupling, some unitary transforms may involve phaseshifts applied to one or more modes. In some photonic implementations,variable phase-shifters may be implemented in integrated circuits,providing control over the relative phases of the state of a photonspread over multiple modes. Examples of transfer matrices that definesuch a phase shifts are given by (for applying a +i and −i phase shiftto the second mode, respectively):

$\begin{matrix}{{s = \begin{pmatrix}1 & 0 \\0 & i\end{pmatrix}}{s^{\dagger} = \begin{pmatrix}1 & 0 \\0 & {- i}\end{pmatrix}}} & (14)\end{matrix}$For silica-on-silicon materials some embodiments implement variablephase-shifters using thermo-optical switches. The thermo-opticalswitches use resistive elements fabricated on the surface of the chip,that via the thermo-optical effect may provide a change of therefractive index n by raising the temperature of the waveguide by anamount of the order of 10⁻⁵ K. One of skill in the art with access tothe present disclosure will understand that any effect that changes therefractive index of a portion of the waveguide may be used to generate avariable, electrically tunable, phase shift. For example, someembodiments use beam splitters based on any material that supports anelectro-optic effect, so-called χ² and χ³ materials such as lithiumniobite, BBO, KTP, and the like and even doped semiconductors such assilicon, germanium, and the like.

Beam-splitters with variable transmissivity and arbitrary phaserelationships between output modes may also be achieved by combiningdirectional couplers and variable phase-shifters in a Mach-ZehnderInterferometer (MZI) configuration 300, e.g., as shown in FIG. 3A.Complete control over the relative phase and amplitude of the two modes302 a, 302 b in dual rail encoding may be achieved by varying the phasesimparted by phase shifters 306 a, 306 b, and 306 c and the length andproximity of coupling regions 304 a and 304 b. FIG. 3B shows a slightlysimpler example of a MZI 310 that allows for a variable transmissivitybetween modes 302 a, 302 b by varying the phase imparted by the phaseshifter 306. FIGS. 3A and 3B are examples of how one could implement amode coupler in a physical device, but any type of mode coupler/beamsplitter may be used without departing from the scope of the presentdisclosure.

In some embodiments, beam splitters and phase shifters may be employedin combination to implement a variety of transfer matrices. For example,FIG. 4A shows, in a schematic form similar to that of FIG. 2A, a modecoupler 400 implementing the following transfer matrix:

$\begin{matrix}{T_{r} = {\frac{1}{\sqrt{2}}{\begin{pmatrix}1 & 1 \\1 & {- 1}\end{pmatrix}.}}} & (15)\end{matrix}$Thus, mode coupler 400 applies the following mappings:

$\begin{matrix}{\left. \left. 10 \right\rangle\mapsto\frac{\left. 10 \right\rangle + \left. 01 \right\rangle}{\sqrt{2}} \right.\left. \left. 01 \right\rangle\mapsto\frac{\left. 10 \right\rangle - \left. 01 \right\rangle}{\sqrt{2}} \right.\left. \left. 11 \right\rangle\mapsto{{1/2}{\left( {\left. 20 \right\rangle + \left. 02 \right\rangle} \right).}} \right.} & (16)\end{matrix}$The transfer matrix T_(r) of Eq. (15) is related to the transfer matrixT of Eq. (9) by a phase shift on the second mode. This is schematicallyillustrated in FIG. 4A by the closed node 407 where mode coupler 416couples to the first mode (line 212) and open node 408 where modecoupler 416 couples to the second mode (line 214). More specifically,T_(r)=sTs, and, as shown at the right-hand side of FIG. 4A, mode coupler416 may be implemented using mode coupler 216 (as described above), witha preceding and following phase shift (denoted by open squares 418 a,418 b). Thus, the transfer matrix T_(r) may be implemented by thephysical beam splitter shown in FIG. 4B, where the open trianglesrepresent +i phase shifters.

Similarly, networks of mode couplers and phase shifters may be used toimplement couplings among more than two modes. For example, FIG. 5 showsa four-mode coupling scheme that implements a “spreader,” or“mode-information erasure,” transformation on four modes, i.e., it takesa photon in any one of the input modes and delocalizes the photonamongst each of the four output modes such that the photon has equalprobability of being detected in any one of the four output modes. (Thewell-known Hadamard transformation is one example of a spreadertransformation.) As in FIG. 2A, the horizontal lines 512-515 correspondto modes, and the mode coupling is indicated by a vertical line 516 withnodes (dots) to identify the modes being coupled. In this case, fourmodes are coupled. Circuit notation 502 is an equivalent representationto circuit diagram 504, which is a network of first-order modecouplings. More generally, where a higher-order mode coupling may beimplemented as a network of first-order mode couplings, a circuitnotation similar to notation 502 (with an appropriate number of modes)may be used.

FIG. 6 illustrates an example optical device 600 that may implement thefour-mode mode-spreading transform shown schematically in FIG. 5 inaccordance with some embodiments. Optical device 600 includes a firstset of optical waveguides 601, 603 formed in a first layer of material(represented by solid lines in FIG. 6) and a second set of opticalwaveguides 605, 607 formed in a second layer of material that isdistinct and separate from the first layer of material (represented bydashed lines in FIG. 6). The second layer of material and the firstlayer of material are located at different heights on a substrate. Oneof ordinary skill will appreciate that an interferometer such as thatshown in FIG. 6 could be implemented in a single layer if appropriatelow loss waveguide crossing were employed.

At least one optical waveguide 601, 603 of the first set of opticalwaveguides is coupled with an optical waveguide 605, 607 of the secondset of optical waveguides with any type of suitable optical coupler,e.g., the directional couplers described herein (e.g., the opticalcouplers shown in FIGS. 2B, 3A, 3B). For example, the optical deviceshown in FIG. 6 includes four optical couplers 618, 620, 622, and 624.Each optical coupler may have a coupling region in which two waveguidespropagate in parallel. Although the two waveguides are illustrated inFIG. 6 as being offset from each other in the coupling region, the twowaveguides may be positioned directly above and below each other in thecoupling region without offset. In some embodiments, one or more of theoptical couplers 618, 620, 622, and 624 are configured to have acoupling efficiency of approximately 50% between the two waveguides(e.g., a coupling efficiency between 49% and 51%, a coupling efficiencybetween 49.9% and 50.1%, a coupling efficiency between 49.99% and50.01%, or a coupling efficiency of 50%, etc.). For example, the lengthof the two waveguides, the refractive indices of the two waveguides, thewidths and heights of the two waveguides, the refractive index of thematerial located between two waveguides, and the distance between thetwo waveguides are selected to provide the coupling efficiency of 50%between the two waveguides. This allows the optical coupler to operatelike a 50/50 beam splitter.

In addition, the optical device shown in FIG. 6 may include twointer-layer optical couplers 614 and 616. Optical coupler 614 allowstransfer of light propagating in a waveguide on the first layer ofmaterial to a waveguide on the second layer of material, and opticalcoupler 616 allows transfer of light propagating in a waveguide on thesecond layer of material to a waveguide on the first layer of material.The optical couplers 614 and 616 allow optical waveguides located in atleast two different layers to be used in a multi-channel opticalcoupler, which, in turn, enables a compact multi-channel opticalcoupler.

Furthermore, the optical device shown in FIG. 6 includes a non-couplingwaveguide crossing region 626. In some implementations, the twowaveguides (603 and 605 in this example) cross each other without havinga parallel coupling region present at the crossing in the non-couplingwaveguide crossing region 626 (e.g., the waveguides may be two straightwaveguides that cross each other at a nearly 90-degree angle).

Those skilled in the art will understand that the foregoing examples areillustrative and that photonic circuits using beam splitters and/orphase shifters may be used to implement many different transfermatrices, including transfer matrices for real and imaginary Hadamardtransforms of any order, discrete Fourier transforms, and the like. Oneclass of photonic circuits, referred to herein as “spreader” or“mode-information erasure (MIE)” circuits, has the property that if theinput is a single photon localized in one input mode, the circuitdelocalizes the photon amongst each of a number of output modes suchthat the photon has equal probability of being detected in any one ofthe output modes. Examples of spreader or MIE circuits include circuitsimplementing Hadamard transfer matrices. (It is to be understood thatspreader or MIE circuits may receive an input that is not a singlephoton localized in one input mode, and the behavior of the circuit insuch cases depends on the particular transfer matrix implemented.) Inother instances, photonic circuits may implement other transfermatrices, including transfer matrices that, for a single photon in oneinput mode, provide unequal probability of detecting the photon indifferent output modes.

In some embodiments, entangled states of multiple photonic qubits may becreated by coupling modes of two (or more) qubits and performingmeasurements on other modes. By way of example, FIG. 7 shows a circuitdiagram for a Bell state generator 700 that may be used in somedual-rail-encoded photonic embodiments. In this example, modes732(1)-732(4) are initially each occupied by a photon (indicated by awavy line); modes 732(5)-732(8) are initially vacuum modes. (Thoseskilled in the art will appreciate that other combinations of occupiedand unoccupied modes may be used.)

A first-order mode coupling (e.g., implementing transfer matrix T of Eq.(9)) is performed on pairs of occupied and unoccupied modes as shown bymode couplers 731(1)-731(4). Thereafter, a mode-information erasurecoupling (e.g., implementing a four-mode mode spreading transform asshown in FIG. 5) is performed on four of the modes (modes732(5)-732(8)), as shown by mode coupler 737. Modes 732(5)-732(8) act as“heralding” modes that are measured and used to determine whether a Bellstate was successfully generated on the other four modes 732(1)-732(4).For instance, detectors 738(1)-738(4) may be coupled to the modes732(5)-732(8) after second-order mode coupler 737. Each detector738(1)-738(4) may output a classical data signal (e.g., a voltage levelon a conductor) indicating whether it detected a photon (or the numberof photons detected). These outputs may be coupled to classical decisionlogic circuit 740, which determines whether a Bell state is present onthe other four modes 732(1)-732(4). For example, decision logic circuit740 may be configured such that a Bell state is confirmed (also referredto as “success” of the Bell state generator) if and only if a singlephoton was detected by each of exactly two of detectors 738(1)-738(4).Modes 732(1)-732(4) may be mapped to the logical states of two qubits(Qubit 1 and Qubit 2), as indicated in FIG. 7. Specifically, in thisexample, the logical state of Qubit 1 is based on occupancy of modes732(1) and 732(2), and the logical state of Qubit 2 is based onoccupancy of modes 732(3) and 732(4). It should be noted that theoperation of Bell state generator 700 may be non-deterministic; that is,inputting four photons as shown does not guarantee that a Bell statewill be created on modes 732(1)-732(4). In one implementation, theprobability of success is 4/32.

In some embodiments, it is desirable to form cluster states of multipleentangled qubits (typically 3 or more qubits, although the Bell statemay be understood as a cluster state of two qubits). One technique forforming larger entangled systems is through the use of an entanglingmeasurement, which is a projective measurement that may be employed tocreate entanglement between systems of qubits. As used herein, “fusion”(or “fusion operation”, “fusion measurement”, or “fusing”) refers to atwo-qubit entangling measurement. A “fusion gate” is a structure thatreceives two input qubits, each of which is typically part of anentangled system. The fusion gate performs a projective measurementoperation on the input qubits that produces either one (“type I fusion”)or zero (“type II fusion”) output qubits in a manner such that theinitial two separate entangled systems are fused into a single entangledsystem. Fusion gates are specific examples of a general class oftwo-qubit entangling measurements and are particularly suited forphotonic architectures. Type II fusion may more generally referred to as“Bell fusion”. In general, a fusion measurement is a projectiveentangling measurement on multiple qubits, which is implemented by afusion device receiving input qubits and outputting classical bitsgiving measurement outcomes.

In a photonic implementation (e.g., one that relies on linear optics),the Bell fusion is probabilistic. It may be implemented on dual-railphotonic qubits using a linear optic circuit. When two qubits fromdifferent entangled stabilizer states undergo type-II fusion, withprobability (1-p_(fail)), the fusion “succeeds” and measures the inputqubits in the Bell stabilizer basis X₁X₂, Z₁Z₂ as intended. However,with probability p_(fail), the fusion “fails” and performs separablesingle qubit measurements Z₁I₂,I₁Z₂ instead.

The failure of type-II fusion is a more benign error than erasure sinceit is heralded and does not create a mixed state. Pure stabilizermeasurements may be obtained regardless of whether there is a success orfailure outcome of the fusion. Even in the case of failure, the outcomeZ₁Z₂ may be obtained by multiplying the two single qubit measurementstogether, and this allows utilization of the same measurement basis inthe case of both success and failure. For example, fusion failure may betreated as a Bell measurement with the information for the X₁X₂measurement erased. In some embodiments, a type-II fusion involves twobeamsplitters and four detectors (e.g., as shown in FIG. 9A), and has afailure probability of 50%.

Examples of type I and type II fusion gates will now be described.

FIG. 8A shows a circuit diagram illustrating a type I fusion gate 800 inaccordance with some embodiments. The diagram shown in FIG. 8A isschematic with each horizontal line representing a mode of a quantumsystem, e.g., a waveguide that may potentially be occupied by one ormore photons. In dual-rail encoding, each pair of modes represents aqubit. In a photonic implementation of the gate the modes in diagramssuch as that shown in FIG. 8A may be physically realized using singlephotons in photonic waveguides. More generally, a type I fusion gatelike that shown in FIG. 8A takes qubit A (physically realized, e.g., byphoton modes 843 and 845) and qubit B (physically realized, e.g., byphoton modes 847 and 849) as input and outputs a single “fused” qubitthat inherits the entanglement with other qubits that were previouslyentangled with either (or both) of input qubit A or input qubit B.

For example, FIG. 8B shows the result of type-I fusing of two qubits Aand B that are each, respectively, a qubit located at the end (i.e., aleaf) of some longer entangled cluster state (only a portion of which isshown). The qubit 857 that remains after the fusion operation inheritsthe entangling bonds from the original qubits A and B thereby creating alarger linear cluster state. FIG. 8B also shows the result of type-Ifusing of two qubits A and B that are each, respectively, an internalqubit that belongs to some longer entangled cluster of qubits (only aportion of which is shown). As before, the qubit 859 that remains afterfusion inherits the entangling bonds from the original qubits A and Bthereby creating a fused cluster state. In this case, the qubit thatremains after the fusion operation is entangled with the larger clusterby way of four other nearest neighbor qubits as shown.

Returning to the schematic illustration of a type I fusion gate 800shown in FIG. 8A, qubit A is dual-rail encoded by modes 843 and 845, andqubit B is dual-rail encoded by modes 847 and 849. For example, in thecase of path-encoded photonic qubits, the logical zero state of qubit A(denoted |0

_(A)) occurs when mode 843 is a photonic waveguide that includes asingle photon and mode 845 is a photonic waveguide that includes zerophotons (and similarly for qubit B). Thus, type I fusion gate 800 maytake as input two dual-rail-encoded photon qubits thereby resulting in atotal of four input modes (e.g., modes 843, 845, 847, and 849). Toaccomplish the fusion operation, a mode coupler (e.g., 50/50 beamsplitter) 853 is applied between a mode of each of the input qubits,e.g., between mode 843 and mode 849 before performing a detectionoperation on both modes using photon detectors 855 (which includes twodistinct photon detectors coupled to modes 843 and 849 respectively). Inaddition, to ensure that the output modes are adjacently positioned, amode swap operation 851 may be applied that swaps the position of thesecond mode of qubit A (mode 845) with the position the second mode ofqubit B (mode 849). In some embodiments, mode swapping may beaccomplished through a physical waveguide crossing as described above orby one or more photonic switches or by any other type of physical modeswap.

FIG. 8A shows only an example arrangement for a type I fusion gate andone of ordinary skill will appreciate that the position of the modecoupler and the presence of the mode swap region 851 may be alteredwithout departing from the scope of the present disclosure. For example,beam splitter 853 may be applied between modes 845 and 847. Mode swapsare optional and are not necessary if qubits having non-adjacent modesmay be dealt with, e.g., by tracking which modes belong to which qubitsby storing this information in a classical memory.

Type I fusion gate 800 is a nondeterministic gate, i.e., the fusionoperation succeeds with a certain probability less than 1, and in othercases the quantum state that results is not a larger cluster state thatcomprises the original cluster states fused together to a larger clusterstate. More specifically, gate 800 “succeeds,” with probability 50%,when only one photon is detected by detectors 855, and “fails” if zeroor two photons are detected by detectors 855. When the gate succeeds,the two cluster states that qubits A and B were a part of become fusedinto a single larger cluster state with a fused qubit remaining as thequbit that links the two previously unlinked cluster states (see, e.g.,FIG. 8B). However, when the fusion gate fails, it has the effect ofremoving both qubits from the original cluster resource states withoutgenerating a larger fused state. A third (less likely) possibility is aloss outcome, where one or more photons within the input states 843-849escape from their respective waveguides.

FIG. 9A shows a circuit diagram illustrating a type II fusion gate 900in accordance with some embodiments. Like other diagrams herein, thediagram shown in FIG. 9A is schematic with each horizontal linerepresenting a mode of a quantum system, e.g., a waveguide that maypotentially be occupied by one or more photons. In a dual-rail encoding,each pair of modes represents a qubit. In a photonic implementation ofthe gate the modes in diagrams such as that shown in FIG. 9A may bephysically realized using single photons in photonic waveguides. Mostgenerally, a type II fusion gate such as gate 900 takes qubit A(physically realized, e.g., by photon modes 943 and 945) and qubit B(physically realized, e.g., by photon modes 947 and 949) as input andoutputs a quantum state that inherits the entanglement with other qubitsthat were previously entangled with either (or both) of input qubit A orinput qubit B. (For type II fusion, if the input quantum state had Nqubits, the output quantum state has N−2 qubits. This is different fromtype I fusion where an input quantum state of N qubits leads to anoutput quantum state having N−1 qubits.)

For example, FIG. 9B shows the result of type-II fusing of two qubits Aand B that are each, respectively, a qubit located at the end (i.e., aleaf) of some longer entangled cluster state (only a portion of which isshown). The resulting qubit system 971 inherits the entangling bondsfrom qubits A and B thereby creating a larger linear cluster state.

Returning to the schematic illustration of type II fusion gate 900 shownin FIG. 9A, qubit A is dual-rail encoded by modes 943 and 945, and qubitB is dual-rail encoded by modes 947 and 949. For example, in the case ofpath encoded photonic qubits, the logical zero state of qubit A (denoted|0

_(A)) occurs when mode 943 is a photonic waveguide that includes asingle photon and mode 945 is a photonic waveguide that includes zerophotons (and likewise for qubit B). Thus, type II fusion gate 900 takesas input two dual-rail-encoded photon qubits thereby resulting in atotal of four input modes (e.g., modes 943, 945, 947, and 949). Toaccomplish the fusion operation, a first mode coupler (e.g., 50/50 beamsplitter) 953 is applied between a mode of each of the input qubits,e.g., between mode 943 and mode 949, and a second mode coupler (e.g.,50/50 beam splitter) 955 is applied between the other modes of each ofthe input qubits, e.g., between modes 945 and 947. A detection operationis performed on all four modes using photon detectors 957(1)-957(4). Insome embodiments, mode swap operations (not shown in FIG. 9A) may beperformed to place modes in adjacent positions prior to mode coupling.In some embodiments, mode swapping may be accomplished through aphysical waveguide crossing as described above or by one or morephotonic switches or by any other type of physical mode swap. Mode swapsare optional and are not necessary if qubits having non-adjacent modesmay be dealt with, e.g., by tracking which modes belong to which qubitsby storing this information in a classical memory.

FIG. 9A shows only an example arrangement for the type II fusion gateand one of ordinary skill will appreciate that the positions of the modecouplers and the presence or absence of mode swap regions may be alteredwithout departing from the scope of the present disclosure.

The type II fusion gate shown in FIG. 9A is a nondeterministic gate,i.e., the fusion operation succeeds with a certain probability less than1, and in other cases the quantum state that results is not a largercluster state that comprises the original cluster states fused togetherto a larger cluster state. More specifically, the gate “succeeds” in thecase where one photon is detected by one of detectors 957(1) and 957(4)and one photon is detected by one of detectors 957(2) and 957(3); inother cases where two photons are detected in different combinations ofdetectors, the gate “fails.” When the gate succeeds, the two clusterstates that qubits A and B were a part of become fused into a singlelarger cluster state; unlike type-I fusion, no fused qubit remains(e.g., compare FIG. 8B and FIG. 9B). When the fusion gate fails, it hasthe effect of removing both qubits from the original cluster resourcestates without generating a larger fused state. As a third possibility,if fewer than two photons are detected, one or both photons may haveescaped, resulting in a loss outcome.

The foregoing description provides an example of how photonic circuitsmay be used to implement physical qubits and operations on physicalqubits using mode coupling between waveguides. In these examples, a pairof modes may be used to represent each physical qubit. Examplesdescribed below may be implemented using similar photonic circuitelements.

It should be understood that all numerical values used herein are forpurposes of illustration and may be varied. In some instances, rangesare specified to provide a sense of scale, but numerical values outsidea disclosed range are not precluded.

It should also be understood that all diagrams herein are intended asschematic. Unless specifically indicated otherwise, the drawings are notintended to imply any particular physical arrangement of the elementsshown therein, or that all elements shown are necessary. Those skilledin the art with access to this disclosure will understand that elementsshown in drawings or otherwise described in this disclosure may bemodified or omitted and that other elements not shown or described maybe added.

This disclosure provides a description of the claimed invention withreference to specific embodiments. Those skilled in the art with accessto this disclosure will appreciate that the embodiments are notexhaustive of the scope of the claimed invention, which extends to allvariations, modifications, and equivalents.

FIG. 10 illustrates an example of a qubit entangling system 1001 inaccordance with some embodiments. Such a system may be used to generatequbits (e.g., photons) in an entangled state (e.g., a GHZ state, Bellpair, and the like), in accordance with some embodiments.

In an illustrative photonic architecture, qubit entangling system 1001may include a photon source module 1005 that is optically connected toentangled state generator 1000. Both the photon source module 1005 andthe entangled state generator 1000 may be coupled to a classicalprocessing system 1003 such that the classical processing system 1003may communicate and/or control (e.g., via the classical informationchannels 1030 a-b) the photon source module 1005 and/or the entangledstate generator 1000. Photon source module 1005 may include a collectionof single-photon sources that may provide output photons to entangledstate generator 1000 by way of interconnecting waveguides 1032.Entangled state generator 1000 may receive the output photons andconvert them to one or more entangled photonic states and then outputthese entangled photonic states into output waveguides 1040. In someembodiments, output waveguide 1040 may be coupled to some downstreamcircuit that may use the entangled states for performing a quantumcomputation. For example, the entangled states generated by theentangled state generator 1000 may be used as resources for a downstreamquantum optical circuit (not shown).

In some embodiments, system 1001 may include classical channels 1030(e.g., classical channels 1030-a through 1030-d) for interconnecting andproviding classical information between components. It should be notedthat classical channels 1030-a through 1030-d need not all be the same.For example, classical channel 1030-a through 1030-c may comprise abi-directional communication bus carrying one or more reference signals,e.g., one or more clock signals, one or more control signals, or anyother signal that carries classical information, e.g., heraldingsignals, photon detector readout signals, and the like.

In some embodiments, qubit entangling system 1001 includes the classicalcomputer system 1003 that communicates with and/or controls the photonsource module 1005 and/or the entangled state generator 1000. Forexample, in some embodiments, classical computer system 1003 may be usedto configure one or more circuits, e.g., using system clock that may beprovided to photon sources 1005 and entangled state generator 1000 aswell as any downstream quantum photonic circuits used for performingquantum computation. In some embodiments, the quantum photonic circuitsmay include optical circuits, electrical circuits, or any other types ofcircuits. In some embodiments, classical computer system 1003 includesmemory 1004, one or more processor(s) 1002, a power supply, aninput/output (I/O) subsystem, and a communication bus or interconnectingthese components. The processor(s) 1002 may execute modules, programs,and/or instructions stored in memory 1004 and thereby perform processingoperations.

In some embodiments, memory 1004 stores one or more programs (e.g., setsof instructions) and/or data structures. For example, in someembodiments, entangled state generator 1000 may attempt to produce anentangled state over successive stages, any one of which may besuccessful in producing an entangled state. In some embodiments, memory1004 stores one or more programs for determining whether a respectivestage was successful and configuring the entangled state generator 1000accordingly (e.g., by configuring entangled state generator 1000 toswitch the photons to an output if the stage was successful, or pass thephotons to the next stage of the entangled state generator 1000 if thestage was not yet successful). To that end, in some embodiments, memory1004 stores detection patterns (described below) from which theclassical computing system 1003 may determine whether a stage wassuccessful. In addition, memory 1004 may store settings that areprovided to the various configurable components (e.g., switches)described herein that are configured by, e.g., setting one or more phaseshifts for the component.

In some embodiments, some or all of the above-described functions may beimplemented with hardware circuits on photon source module 1005 and/orentangled state generator 1000. For example, in some embodiments, photonsource module 1005 includes one or more controllers 1007-a (e.g., logiccontrollers) (e.g., which may comprise field programmable gate arrays(FPGAs), application specific integrated circuits (ASICS), a “system ona chip” that includes classical processors and memory, or the like). Insome embodiments, controller 1007-a determines whether photon sourcemodule 1005 was successful (e.g., for a given attempt on a given clockcycle, described below) and outputs a reference signal indicatingwhether photon source module 1005 was successful. For example, in someembodiments, controller 1007-a outputs a logical high value to classicalchannel 1030-a and/or classical channel 1030-c when photon source module1005 is successful and outputs a logical low value to classical channel1030-a and/or classical channel 1030-c when photon source module 1005 isnot successful. In some embodiments, the output of control 1007-a may beused to configure hardware in controller 1007-b.

Similarly, in some embodiments, entangled state generator 1000 includesone or more controllers 1007-b (e.g., logical controllers) (e.g., whichmay comprise field programmable gate arrays (FPGAs), applicationspecific integrated circuits (ASICS), or the like) that determinewhether a respective stage of entangled state generator 1000 hassucceeded, perform the switching logic described above, and output areference signal to classical channels 1030-b and/or 1030-d to informother components as to whether the entangled state generator 400 hassucceeded.

In some embodiments, a system clock signal may be provided to photonsource module 1005 and entangled state generator 1000 via an externalsource (not shown) or by classical computing system 1003 generates viaclassical channels 1030-a and/or 1030-b. In some embodiments, the systemclock signal provided to photon source module 1005 triggers photonsource module 1005 to attempt to output one photon per waveguide. Insome embodiments, the system clock signal provided to entangled stategenerator 1000 triggers, or gates, sets of detectors in entangled stategenerator 1000 to attempt to detect photons. For example, in someembodiments, triggering a set of detectors in entangled state generator1000 to attempt to detect photons includes gating the set of detectors.

It should be noted that, in some embodiments, photon source module 1005and entangled state generator 1000 may have internal clocks. Forexample, photon source module 1005 may have an internal clock generatedand/or used by controller 1007-a and entangled state generator 1000 hasan internal clock generated and/or used by controller 1007-b. In someembodiments, the internal clock of photon source module 1005 and/orentangled state generator 1000 is synchronized to an external clock(e.g., the system clock provided by classical computer system 1003)(e.g., through a phase-locked loop). In some embodiments, any of theinternal clocks may themselves be used as the system clock, e.g., aninternal clock of the photon source may be distributed to othercomponents in the system and used as the master/system clock.

In some embodiments, photon source module 1005 includes a plurality ofprobabilistic photon sources that may be spatially and/or temporallymultiplexed, i.e., a so-called multiplexed single photon source. In oneexample of such a source, the source is driven by a pump, e.g., a lightpulse, that is coupled into an optical resonator that, through somenonlinear process (e.g., spontaneous four wave mixing, second harmonicgeneration, and the like) may generate zero, one, or more photons. Asused herein, the term “attempt” is used to refer to the act of driving aphoton source with some sort of driving signal, e.g., a pump pulse, thatmay produce output photons non-deterministically (i.e., in response tothe driving signal, the probability that the photon source will generateone or more photons may be less than 1). In some embodiments, arespective photon source may be most likely to, on a respective attempt,produce zero photons (e.g., there may be a 90% probability of producingzero photons per attempt to produce a single-photon). The second mostlikely result for an attempt may be production of a single-photon (e.g.,there may be a 9% probability of producing a single-photon per attemptto produce a single-photon). The third most likely result for an attemptmay be production of two photons (e.g., there may be an approximately 1%probability of producing two photons per attempt to produce a singlephoton). In some circumstances, there may be less than a 1% probabilityof producing more than two photons.

In some embodiments, the apparent efficiency of the photon sources maybe increased by using a plurality of single-photon sources andmultiplexing the outputs of the plurality of photon sources.

The precise type of photon source used is not critical and any type ofsource may be used, employing any photon generating process, such asspontaneous four wave mixing (SPFW), spontaneous parametricdown-conversion (SPDC), or any other process. Other classes of sourcesthat do not necessarily require a nonlinear material may also beemployed, such as those that employ atomic and/or artificial atomicsystems, e.g., quantum dot sources, color centers in crystals, and thelike. In some cases, sources may or may be coupled to photonic cavities,e.g., as may be the case for artificial atomic systems such as quantumdots coupled to cavities. Other types of photon sources also exist forSPWM and SPDC, such as optomechanical systems and the like. In someexamples the photon sources may emit multiple photons already in anentangled state in which case the entangled state generator 1000 may notbe necessary, or alternatively may take the entangled states as inputand generate even larger entangled states.

For the sake of illustration, an example which employs spatialmultiplexing of several non-deterministic is described as an example ofa MUX photon source. However, many different spatial MUX architecturesare possible without departing from the scope of the present disclosure.Temporal MUXing may also be implemented instead of or in combinationwith spatial multiplexing. MUX schemes that employ log-tree, generalizedMach-Zehnder interferometers, multimode interferometers, chainedsources, chained sources with dump-the-pump schemes, asymmetricmulti-crystal single photon sources, or any other type of MUXarchitecture may be used. In some embodiments, the photon source mayemploy a MUX scheme with quantum feedback control and the like.

1.4 Introduction to Fusion Based Quantum Computing

Quantum computation is often considered in the framework of ‘CircuitBased Quantum Computation’ (CBQC) in which operations (or gates) areperformed on physical qubits. Gates may be either single qubit unitaryoperations (rotations), or two qubit entangling operations such as theCNOT gate.

Fusion Based Quantum Computation (FBQC) is another approach toimplementing quantum computation. In the FBQC approach, computationproceeds by first preparing a particular entangled state of many qubits,commonly referred to as a cluster state, and then carrying out a seriesof single qubit measurements to enact the quantum computation. In thisapproach, the choice of single qubit measurements is dictated by thequantum algorithm being run on the quantum computer. In the FBQCapproach, fault tolerance may be achieved by careful design of thecluster state and using the topology of this cluster state to encode alogical qubit that is protected against errors that may occur on any oneof the physical qubits that make up the cluster state. In practice, thevalue of the logical qubit may be determined based on the results of thesingle-particle measurements that are made of the physical qubits thatform the cluster state as the computation proceeds.

However, the generation and maintenance of long-range entanglementacross the cluster state and subsequent storage of large cluster statesmay be a challenge. For example, for a physical implementation of theFBQC approach proposed by Raussendorf et al., a cluster state containingmany thousands, or more, of mutually entangled qubits must be preparedand then stored for some period of time before the single-qubitmeasurements are performed. For example, to generate a cluster staterepresenting a single logical error corrected qubit, each of thecollection of underlying physical qubits is prepared in the |+

state and a controlled-phase gate (CZ) state is applied between eachphysical qubit pair to generate the overall cluster state. Moreexplicitly, a cluster state of highly entangled qubits described by theundirected graph G=(V, E) with V and E denoting the sets of vertices andedges, respectively may be generated as follows: 1) initialize all thephysical qubits in the |+

state, where |+

=(|0

+|1

)/√{square root over (2)}. 2) apply the controlled-phase gate CZ to eachpair (i, j) of qubits. Accordingly, any cluster state, which physicallycorresponds to a large entangled state of physical qubits, may bedescribed as

$\left. \Psi \right\rangle_{graph} = {\prod\limits_{{({i,j})} \in E}{CZ_{i,j}\left.  + \right\rangle^{\otimes {V}}}}$where the CZ_(i,j) is the controlled phase gate operator. Graphically,any cluster state may be represented by a graph that includes verticesthat represent the physical qubits (initialized in the |+

state) and edges that represent entanglement between them (i.e., theapplication of the various CZ gates).

After |Ψ

_(graph) is generated, this large state of mutually entangled qubitsmust be preserved long enough for a stabilizer measurement to beperformed, e.g., by making x measurements on all physical qubits in thebulk of the lattice and z-measurements on the boundary qubits.

FIG. 11A shows one example of a fault tolerant cluster state that may beused in FBQC. Specifically, FIG. 11A illustrates the topological clusterstate introduced by Raussendorf et al. commonly referred to as theRaussendorf lattice. The cluster state is in the form of repeatinglattice cells (e.g., cell 1120, delineated by non-bold lines) withphysical qubits (e.g., physical qubit 1116) arranged on the faces andedges of the cells. Entanglement between the physical qubits isrepresented by edges that connect the physical qubits (e.g., edge 1118,indicated by bold lines). The cluster state shown in FIG. 11A is merelyone example among many and other cluster states may be used withoutdeparting from the scope of the present disclosure. Furthermore, whilethe example shown here is represented in three spatial dimensions, thesame structure may also be obtained from other implementations of codesthat are not based on a purely spatial entangled cluster state, butrather may include both entanglement in 2D space and entanglement intime, e.g., a 2+1D surface code implementation may be used or any otherfoliated code. For such cluster states, all of the quantum gates neededfor fault tolerant quantum computation may be constructed by making aseries of single particle measurements to the physical qubits that makeup the lattice.

Returning to FIG. 11A, a chunk of a Raussendorf lattice is shown. Suchan entangled state may be used to encode one or more logical qubits(i.e. one or more error corrected qubits) using many entangled physicalqubits. The measurement results of the multiple physical qubits, e.g.,physical qubit 1116, may be used for correcting errors and forperforming fault tolerant computations on the logical qubits through theuse of a decoder. One of ordinary skill will appreciate that the numberof physical qubits required to encode a single logical qubit may varydepending on the precise nature of the physical errors, noise, etc.,that are experienced by the physical qubits, but to achieve faulttolerance, all proposals to date utilize at least thousands of physicalqubits to encode a single logical qubit. Generating and maintaining sucha large entangled state of thousands, tens of thousands, or evenmillions of physical qubits remains a key challenge for any practicalimplementation of the FBQC approach.

FIGS. 11B-11C illustrate how the decoding of a logical qubit couldproceed for the case of a cluster state based on the Raussendorflattice. As may be seen in FIG. 11A, the geometry of the cluster stateis related to the geometry of a cubic lattice (e.g., with lattice cell1120) shown superimposed on the cluster state in FIG. 11A. FIG. 11Bshows the measurement results (also superimposed on the cubic lattice)after the state of each physical qubit of the cluster state has beenmeasured, with the measurement results being placed in the formerposition of the physical qubit that was measured (for clarity onlymeasurement results that result from measurements of the surface qubitsare shown).

In some embodiments, a measured qubit state may be represented by anumerical bit value of either 1 or 0 after all qubits have beenmeasured, e.g., in a particular basis such as the x-basis. Asillustrated, qubits may be classified as one of two types, those thatare located on the edges of a unit cell (e.g. edge qubit 1122), andthose that are located on the faces of a unit cell (e.g., face qubit1124). In some cases, a measurement of the qubit may not be obtained, orthe result of the qubit measurement may be invalid (e.g., due to afailure or loss outcome). In these cases, there is no bit value assignedto the location of the corresponding measured qubit, but instead theoutcome is an erasure, illustrated here as thick line 1126, for example.These measurement outcomes that are known to be missing may bereconstructed during the decoding procedure.

To identify errors in the physical qubits, a syndrome graph is generatedfrom the collection of measurement outcomes resulting from themeasurements of the physical qubits. For example, the bit valuesassociated with a plurality of edge qubits may be combined to create asyndrome value associated with an adjacent vertex that results from theintersection of the respective edges, e.g., vertex 1128 as shown in FIG.11B. A set of syndrome values (or “syndromes”), also referred to hereinas parity checks, may be associated with each vertex of the syndromegraph, as shown in FIG. 11C. More specifically, in FIG. 11C, thecomputed values of some of the vertex parity checks of the syndromegraph are shown. Only the 12 syndrome values on the front face of thecluster state of FIG. 11C are illustrated as 0s and 1s. The parity checkvalues may be found by computing the parity of the bit values associatedwith each edge of the syndrome graph incident to the vertex. In someembodiments, a parity computation entails determining whether the sum ofthe edge values is an even or odd integer, with the parity result beingthe result of the sum modulo 2. If no errors have occurred in thequantum state or in the qubit measurement, then all syndrome valuesshould be even (or 0). On the contrary, if an error occurs, it mayresult in some odd (or 1) syndrome values. Only half of the bit valuesfrom the qubit measurements are associated with the syndrome graph shownin FIG. 11C (e.g., the bits aligned with the edges of the unit cells),and this illustrated syndrome graph is referred to herein as the “primalgraph”. There is another syndrome graph that is related to parity checkson all the bit values located on the faces of the unit cells of thelattice shown, and this graph is referred to as the “dual graph”. Thereis generally an equivalent decoding problem on the syndrome values ofthe dual graph.

As mentioned above, the generation and subsequent storage of largecluster states of qubits may be a challenge. However, some embodiments,methods and systems described herein provide for the generation of a setof classical measurement data (e.g., a syndrome graph) that includes thenecessary correlations for performing quantum error correction, withoutthe need to first generate a large entangled state of qubits in an errorcorrecting code. For example, embodiments disclosed herein describedsystems and method whereby two-qubit (i.e., joint) measurements may beperformed on a collection of much smaller entangled states to generate aset of classical data that includes the long-range correlationsnecessary to generate the syndrome graph for a particular chosen clusterstate, without the need to actually generate the cluster state. In otherwords, in some systems and methods described herein, there is only evergenerated a collection of relatively small entangled states (referred toherein as resource states) and these resource states need not be allentangled together to form a new larger entangled state that is aquantum error correcting code (e.g., a topological code).

For example, as will be described in further detail below, in the caseof linear optical quantum computing using a Raussendorf latticestructure, to generate the syndrome graph data, a destructive fusiongate may be applied to a collection of small entangled states (e.g.,4-GHZ states) that are themselves not entangled with each other and thusare never part of a larger Raussendorf lattice. Despite the fact thatthe individual resource states were not mutually entangled prior to thedestructive fusion measurement, the measurement outcomes that resultfrom the fusion measurements generate a syndrome graph that includes allthe necessary correlations to perform quantum error correction. Suchsystems and methods are described in greater detail below and arereferred to herein as Fusion Based Quantum Computing (FBQC).

1.5 Fusion Based Quantum Computing

FIG. 12 shows a hybrid computing system in accordance with one or moreembodiments. The hybrid computing system 1201 includes a user interfacedevice 1203 that is communicatively coupled to a hybrid quantumcomputing (QC) sub-system 1205, described in more detail below in FIG.13. The user interface device 1203 may be any type of user interfacedevice, e.g., a terminal including a display, keyboard, mouse,touchscreen and the like. In addition, the user interface device mayitself be a computer such as a personal computer (PC), laptop, tabletcomputer and the like. In some embodiments, the user interface device1203 provides an interface with which a user may interact with thehybrid QC subsystem 1205. For example, the user interface device 1203may run software, such as a text editor, an interactive developmentenvironment (IDE), command prompt, graphical user interface, and thelike so that the user may program, or otherwise interact with, the QCsubsystem to run one or more quantum algorithms. In other embodiments,the QC subsystem 1205 may be pre-programmed and the user interfacedevice 1203 may simply be an interface where a user may initiate aquantum computation, monitor the progress, and receive results from thehybrid QC subsystem 1205. Hybrid QC subsystem 1205 may further include aclassical computing system 1207 coupled to one or more quantum computingchips 1209. In some examples, the classical computing system 1207 andthe quantum computing chip 1209 may be coupled to other electroniccomponents 1211, e.g., pulsed pump lasers, microwave oscillators, powersupplies, networking hardware, etc. In some embodiments that requirecryogenic operation, the quantum computing system 1209 may be housedwithin a cryostat, e.g., cryostat 1213. In some embodiments, the quantumcomputing chip 1209 may include one or more constituent chips, e.g.,hybrid electronic chip 1215 and integrated photonics chip 1217. Signalsmay be routed on- and off-chip any number of ways, e.g., via opticalinterconnects 1219 and via other electronic interconnects 1221. Inaddition, the hybrid computing system 1201 may employ a quantumcomputing process, e.g., fusion-based quantum computing (FBQC) thatemploys one or more cluster states of qubits as described in furtherdetail herein.

FIG. 13 shows a block diagram of a QC system 1301 in accordance withsome embodiments. Such a system may be associated with the computingsystem 1201 introduced above in reference to FIG. 12. In FIG. 13, solidlines represent quantum information channels and dashed lines representclassical information channels. The QC system 1301 includes a qubitentangling system 1303, qubit fusion system 1305, and classicalcomputing system 1307. In some embodiments, the qubit entangling system1303 takes as input a collection of N physical qubits, e.g., physicalqubits 1309 (also represented schematically as inputs 1311 a, 1311 b,1311 c, . . . , 1311N) and generates quantum entanglement between two ormore of them to generate entangled resource states 1315. For example, inthe case of photonic qubits, the qubit entangling system 1303 may be alinear optical system such as an integrated photonic circuit thatincludes waveguides, beam splitters, photon detectors, delay lines, andthe like. In some examples, the entangled resource states 1315 may berelatively small entangled states of qubits, (e.g., Bell states of twoqubits, 3-GHZ states, 4-GHZ states, etc.) or may be small entangledstates that are not large enough to operate as a quantum errorcorrecting code. In some embodiments, the resource states are chosensuch that the fusion operations applied to certain qubits of thesestates results in syndrome lattice data that includes the requiredcorrelations for quantum error correction. Advantageously, the systemshown in FIG. 13 provides for fault tolerant quantum computation usingrelatively small resource states, without requiring that the resourcestates become mutually entangled with each other to form a lattice,cluster, or graph state prior to the final measurements.

In some embodiment, the input qubits 1309 may be a collection of quantumsystems and/or particles and may be formed using any qubit architecture.For example, the quantum systems may be particles such as atoms, ions,nuclei, and/or photons. In other examples, the quantum systems may beother engineered quantum systems such as flux qubits, phase qubits, orcharge qubits (e.g., formed from a superconducting Josephson junction),topological qubits (e.g., Majorana fermions), or spin qubits formed fromvacancy centers (e.g., nitrogen vacancies in diamond). Furthermore, forthe sake of clarity of description, the term “qubit” is used hereinalthough the system may also employ quantum information carriers thatencode information in a manner that is not necessarily associated with abinary bit. For example, qudits (i.e., quantum systems that encodeinformation in more than two quantum states) may be used in accordancewith some embodiments.

In accordance with some embodiments, the QC system 1301 may be afusion-based quantum computer. For example, a software program (e.g., aset of machine-readable instructions) that represents the quantumalgorithm to be run on the QC system 1301 may be passed to a classicalcomputing system 1307 (e.g., corresponding to system 1208 in FIG. 12above). The classical computing system 1307 may be any type of compingdevice such as a PC, one or more blade servers, and the like, or even ahigh-performance computing system such as a supercomputer, server farm,and the like. Such a system may include one or more processors (notshown) coupled to one or more computer memories, e.g., memory 1306. Sucha computing system will be referred to herein as a “classical computer.”In some examples, the software program may be received by a classicalcomputing module, referred to herein as a fusion pattern generator 1313.One function of the fusion pattern generator 1313 is to generate a setof machine-level instructions from the input software program (which mayoriginate as code that may be more easily written by a user to programthe quantum computer). In some embodiments, the fusion pattern generator1313 operates as a compiler for software programs to be run on thequantum computer. Fusion pattern generator 1313 may be implemented aspure hardware, pure software, or any combination of one or more hardwareor software components or modules. In various embodiments, fusionpattern generator 1313 may operate at runtime or in advance; in eithercase, machine-level instructions generated by fusion pattern generator1313 may be stored (e.g., in memory 1306). In some examples, thecompiled machine-level instructions take the form of one or more dataframes that instruct, for a given clock cycle, the qubit fusion system1305 to make one or more fusions between certain qubits from theseparate, mutually unentangled resource states 1315. For example, fusionpattern data frame 1317 is one example of the set of fusion measurements(e.g., type two fusion measurements) that should be applied betweencertain pairs of qubits from different entangled resource states 1315during a certain clock cycle as the program is executed. In someembodiments, several fusion pattern data frames 1317 may be stored inmemory 1306 as classical data. In some embodiments, the fusion patterndata frames 1317 may dictate whether or not XX Type II Fusion is to beapplied (or whether any other type of fusion, or not, is to be applied)for a particular fusion gate within the fusion array 1321 of the qubitfusion system 1305. In addition, the fusion pattern data frames 1317 mayindicate that the Type II fusion is to be performed in a differentbasis, e.g., XX, XY, ZZ, etc.

A fusion controller circuit 1319 of the qubit fusion system 1205 mayreceive data that encodes the fusion pattern data frames 1317 and, basedon this data, may generate configuration signals, e.g., analog and/ordigital electronic signals, that drive the hardware within the fusionarray 1321. For example, for the case of photonic qubits, the fusiongates may include photon detectors coupled to one or more waveguides,beam splitters, interferometers, switches, polarizers, polarizationrotators and the like. More generally, the detectors may be any detectorthat can detect the quantum states of one or more of the qubits in theresource states 1315. One of ordinary skill will appreciate that manytypes of detectors may be used depending on the particular qubitarchitecture being employed

In some embodiments, the result of applying the fusion pattern dataframes 1317 to the fusion array 1321 is the generation of classical data(generated by the fusion gates' detectors) that is read out, andoptionally pre-processed, and sent to decoder 1333. More specifically,the fusion array 1321 may include a collection of measuring devices thatimplement the joint measurements between certain qubits from twodifferent resource states and generate a collection of measurementoutcomes associated with the joint measurement. These measurementoutcomes may be stored in a measurement outcome data frame, e.g., dataframe 1322 and passed back to the classical computing system for furtherprocessing.

In some embodiments, any of the submodules in the QC system 1301, e.g.,controller 1323, quantum gate array 1325, fusion array 1321, fusioncontroller 1319, fusion pattern generator 1313, decoder 1323, andlogical processor 1308 may include any number of classical computingcomponents such as processors (CPUs, GPUs, TPUs) memory (any form ofRAM, ROM), hard coded logic components (classical logic gates such asAND, OR, XOR, etc.) and/or programmable logic components such as fieldprogrammable gate arrays (FPGAs and the like). These modules may alsoinclude any number of application specific integrated circuits (ASICs),microcontrollers (MCUs), systems on a chip (SOCs), and other similarmicroelectronics.

In some embodiments, the entangled resource states 1315 may be any typeof entangled resource state, that, when the fusion operations areperformed, produces measurement outcome data frames that include thenecessary correlations for performing fault tolerant quantumcomputation. While FIG. 13 shows an example of a collection of identicalresource states, a system may be employed that generates many differenttypes of resource states and may even dynamically change the type ofresource state being generated based on the demands of the quantumalgorithm being run. As described herein, the logical qubit measurementoutcomes 1327 may be fault tolerantly recovered, e.g., via decoder 1333,from the measurement outcomes 1322 of the physical qubits. Logicalprocessor 1308 may then process the logical outcomes as part of therunning of the program. As shown, the logical processor may feed backinformation to the fusion pattern generator 1313 for affect downstreamgates and/or measurements to ensure that the computation proceeds faulttolerantly.

FIG. 14 shows one example of qubit fusion system 1401 in accordance withsome embodiments. In some embodiments, qubit fusion system 1401 may beemployed within a larger FBQC system such as qubit fusion system 1305shown in FIG. 13.

Qubit fusion system 1401 includes a fusion controller 1419 that iscoupled to a fusion array 1421. Fusion controller 1419 is configured tooperate as described above in reference to fusion controller circuit1319 of FIG. 13 above. Fusion array 1421 includes a collection of fusionsites that each receive two or more qubits from different resourcestates (not shown) and perform one or more fusion operations (e.g., TypeII fusion) on selected qubits from the two or more resource states. Thefusion operations performed on the qubits may be controlled by thefusion controller 1419 via signals that are sent from the fusioncontroller 1419 to each of the fusion gates via control channels 1403 a,1403 b, etc. Based on the joint measurements performed at each fusionsite, classical measurement outcomes in the form of classical data areoutput and then provided to a decoder system, as shown and describedabove in reference to FIG. 13. Examples of photonic circuits that may beemployed as Type II fusion gates are described below.

FIG. 15 shows one possible example of a fusion site 1501 as configuredto operate with a fusion controller 1319 to provide measurement outcomesto a decoder for fault tolerant quantum computation in accordance withsome embodiments. In this example, fusion site 1501 may be an element offusion array 1321 (shown in FIG. 13), and although only one instance isshown for purposes of illustration, fusion array 1321 may include anynumber of instances of fusion sites 1501.

As described above, the qubit fusion system 1305 may receive two or morequbits (Qubit 1 and Qubit 2) that are to be fused. Qubit 1 is one qubitthat is entangled with one or more other qubits (not shown) as part of afirst resource state and Qubit 2 is another qubit that is entangled withone or more other qubits (not shown) as part of a second resource state.Advantageously, in contrast to ordinary fusion-based quantum computing,none of the qubits from the first resource state need be entangled withany of the qubits from the second (or any other) resource state in orderto facilitate a fault tolerant quantum computation. Also advantageously,at the inputs of a fusion site 1501, the collection of resource statesare not mutually entangled to form a cluster state that takes the formof a quantum error correcting code and thus there is no need to storeand or maintain a large cluster state with long-range entanglementacross the entire cluster state. Also advantageously, the fusionoperations that take place at the fusion sites are fully destructivejoint measurements between Qubit 1 and Qubit 2 such that all that isleft after the measurement is classical information representing themeasurement outcomes on the detectors, e.g., detectors 1503, 1505, 1507,1509. At this point, the classical information is all that is needed forthe decoder 333 to perform quantum error correction, and no furtherquantum information is propagated through the system. This may becontrasted with an FBQC system that might employ fusion sites to fuseresource states into a cluster state that serves as a topological codeand only then generates the required classical information via singleparticle measurements on each qubit in the large cluster state. In suchan FBQC system, not only does the large cluster state need to be storedand maintained but an extra single particle measurement step needs to beapplied, in addition to the fusions used to generate the cluster state,to every qubit of the cluster state in order to generate the classicalinformation necessary for the decoder to perform quantum errorcorrection.

FIG. 15 shows an illustrative example for one way to implement a fusionsite as part of a photonic quantum computer architecture, according tosome embodiments. In this example, qubit 1 and qubit 2 may be dual railencoded photonic qubits. Accordingly, qubit 1 and qubit 2 are input onwaveguides 1521, 1523 and 1525, 1527, respectively. An interferometer1524, 1528 may be placed in line with each qubit, and within one arm ofeach interferometer 1524, 1528 a programmable phase shifter 1530, 1532may be applied to affect the basis in which the fusion operation isapplied, e.g., XX, XY, YY, ZZ, etc.). The programmable phase shifters1530, 1532 may be coupled to the fusion controller 1319 via control line1529 and 1531 such that signals from the fusion controller 1319 may beused to set the basis in which the fusion operation is applied to thequbits. For example, the programmable phase shifters may be programmableto either apply or not apply a Hadamard gate to their respective qubits,altering the basis (e.g., x vs. z) of the type II fusion measurement. Insome embodiments the basis may be hard-coded within the fusioncontroller 1319, or in some embodiments the basis may be chosen basedupon external inputs, e.g., instructions provided by the fusion patterngenerator 1313. In particular, in some embodiments, adaptive basisselection may be employed, wherein the basis of each fusion measurementis selected based on the results of one or more previous fusionmeasurements. Additional mode couplers, e.g., mode couplers 1533 and1532 may be applied after the interferometers followed by single photondetectors 1503, 1505, 1507, 1509 to provide a readout mechanism forperforming the joint measurement. In the example shown in FIG. 15, thefusion site implements an un-boosted Type II fusion operation on theincoming qubits. One of ordinary skill will appreciate that any type offusion operation may be applied (and may be boosted or un-boosted)without departing from the scope of the present disclosure. In someembodiments, the fusion controller 1319 may also provide a controlsignal to the detectors 1503, 1505, 1507, 1509. A control signal may beused, e.g., for gating the detectors or for otherwise controlling theoperation of the detectors. Each of the detectors 1503, 1505, 1507, 1509provides one bit of information (representing a “photon detected” or “nophoton detected” state of the detector), and these four bits may bepreprocessed at the fusion site 1501 to determine a measurement outcome(e.g., fusion success or not) or passed directly to the decoder 1333 forfurther processing.

Success, Failure, and Loss Measurement Outcomes

Several different measurement outcomes may result from a fusionmeasurement. As one example, a fusion measurement may be a jointmeasurement on qubit 1 and qubit 2 to measure a) the product of theirqubit values in the x-basis (i.e., X₁X₂), and b) the product of theirqubit values in the z-basis (i.e., Z₁Z₂). Two respective edges of theprimal and dual syndrome graphs may correspond to each of these jointmeasurement results.

As a first possibility, if the fusion measurement is a “success”, thenboth of these values will be returned as a result of the dual-qubitfusion measurement. However, even in the absence of photon loss, linearoptic fusion does not always produce this result.

As a second possibility (termed herein a “failure”), instead ofmeasuring X₁X₂ and Z₁Z₂, the two-qubit fusion measurement may measure Z₁and Z₂. In this case, the measurement outcome may be used to deduce Z₁Z₂(e.g., by multiplying Z₁ and Z₂), but may not be able to deduce X₁X₂(e.g., because a definite value of Z₁ and Z₂ may correspond to either ofa positive or negative value for the product X₁X₂). Accordingly, theedge in one of the graphs (e.g., the primal graph) corresponding to theZ₁Z₂ measurement will be successfully measured, while the edge in theother graph (e.g., the dual graph) corresponding to the X₁X₂ measurementwill be erased. In this example, the fusion measurement is said to beperformed with a failure basis of X₁X₂. In other words, the “failurebasis” refers to the measurement outcome that risks erasure for aparticular fusion measurement. In general, each of the primal and dualgraphs contain edges corresponding to both Z₁Z₂ and X₁X₂ measurements,and a successful Z₁Z₂ measurement may measure either a primal or dualgraph edge, depending on the particular edge being measured. If aHadamard gate is applied to the dual-rail photonic system, themeasurement basis may be altered such that X₁ and X₂ are measured in afailure outcome (i.e., rather than Z₁ and Z₂). In this case, X₁ and X₂may be used to deduce the product X₁X₂, but the edge corresponding tothe Z₁Z₂ measurement will be erased.

As a third possibility (termed herein a “loss”), one or both of thedual-rail photons may tunnel out and not be detected at all. In thiscase, since one or both of the dual-rail photons is missing, neitherX₁X₂ or Z₁Z₂ will be measured, and both edges will be erased.

1.5.1 Syndrome Graphs in FBQC

FIGS. 16A-E illustrate various aspects and elements of syndrome graphsin an FBQC scheme for fault tolerant quantum computation, in accordancewith one or more embodiments. In the examples shown in FIGS. 16A-E, atopological code utilizing 6-qubit cluster states is used. However, moregenerally any other error correcting code may be used without departingfrom the scope of the present disclosure. FBQC may be implemented usingany type of code, e.g., various volume codes (such as the diamond code,triamond code, etc.), and various color codes, or other topologicalcodes may be used without departing from the scope of the presentdisclosure.

In FBQC, through a series of joint measurements on two or more qubits(e.g., measurements of a positive-operator valued measure, also referredto as a POVM), a set of classical data may be generated that correspondsto the error syndrome of a quantum error correcting code. FIG. 16Aillustrates two 6-qubit resource states within a unit cell of a lattice,where the circles represent individual qubits, the lines connecting thequbits represent entanglement between respective qubits, and the dashedlines delineate the boundary of the unit cell. Using a plurality of unitcells with resource states as illustrated in FIG. 16A, a set ofmeasurements that may be used to generate a syndrome graph value isshown in FIG. 16B. As illustrated, the indicated sets of 6 qubits areused as resource states and the illustrated pairings between neighboringqubits of two different resource states represent fusion measurements onthe respective neighboring qubits. FIG. 16B illustrates resource statesand fusion measurements for several unit cells, where the latticeextends in three dimensions. For clarity, only one repetition isillustrated in the dimension coming out of the page, with a subset ofthe fusions along that dimension shown. However, FIG. 16B isillustrative only, and the actual lattice implemented according toembodiments herein may extend significantly farther in one, two, orthree dimensions. For example, typically each of the 6 qubits of each ofthe plurality of resource states will be involved in a fusionmeasurement. However, for clarity, only those fusion measurements areshown where both of the involved qubits are visible in the truncatedlattice illustrated in FIG. 16B. In order to generate a desired errorsyndrome, a lattice preparation protocol (LPP) is chosen that generatesthe appropriate syndrome lattice from the fusions of the multiplesmaller entangled states, referred to herein as resource states.

FIG. 16C illustrates two resource states in a unit cell (solid lines)with the primal and dual syndrome graph edges indicated with shortdotted lines and long dotted lines, respectively. The dual syndromegraph is geometrically identical to the primal syndrome graph, exceptthat it is shifted in the three Cartesian directions by half of the unitcell distance, relative to the primal syndrome graph. FIG. 16Dadditionally illustrates correspondence between several primal and dualgraph edges. Primal and dual measurements coming from the same fusionmeasurement have the same midpoint and are paired, and several examplesof pairings between primal and dual syndrome graph edges are illustratednumerically in FIG. 16D (e.g., (1,1), (2,2) and (3,3)).

In some embodiments, the fusion measurements on pairs of qubits aresequentially executed in a predetermined fusion measurement sequenceuntil each qubit pair has been measured. Similar to the descriptionabove in reference to FIGS. 11A-C, the results of each fusionmeasurement may be used to populate values in each of a primal and dualsyndrome graph corresponding to the set of resource states. FIG. 16Eillustrates several unit cells of the primal syndrome graph (left) anddual syndrome graph (right) that correspond to the resource statesillustrated in Figure B. Note that the dual syndrome graph in FIG. 16Eis rotated relative to the primal syndrome graph for ease ofvisualization. Further note that, in these embodiments, the syndromegraph is self-dual such that the primal and dual syndrome graphs havethe same topological structure. For each sequential fusion measurement,an edge value will be determined for and edge in each of the primal anddual syndrome graphs. Depending on the result of the fusion measurement,the edge value may be either successfully measured or erased. To avoidan overall logical error in the logical qubit encoded by the set ofresource states, it is desirable to avoid a contiguous chain of erasededges that spans the overall lattice, in either the primal or dualsyndrome graph. Embodiments herein present methods and devices forselectively adjusting the basis for performing the fusion measurementsto reduce the likelihood of obtaining such a lattice-spanning chain oferasures (and thereby obtaining an overall logical error).

As used herein, the term “edge” is intended to refer to an edge ineither the primal syndrome graph or the dual syndrome graph. An edge mayhave one of three edge values: it may be unmeasured, successfullymeasured, or erased. An erased edge may result from either a failedmeasurement or a photon loss, as explained in greater detail below.

1.5.2 FIGS. 17A-D—Connected Components—2D Example

FIG. 17A-D are simplified diagrams of 2-dimensional syndrome graphsillustrating the relationship between the primal and dual graphs duringa sequence of fusion measurements. Actual implementations of FBQC willtypically employ 3-dimensional syndrome graphs. However, the essentialconcepts of primal/dual correspondence and connected components may bemore easily visualized and explained in the context of a 2-dimensionalsyndrome graph. It may be appreciated by one of skill in the art thatthe concepts introduced herein for a connected component in the contextof a 2D syndrome graph may be generalized to apply to a 3D syndromegraph.

FIG. 17A illustrates an overlay of an example 2D primal syndrome graph(dotted lines) and a 2D dual syndrome graph (dashed lines). Solidcircles connected to dotted lines are syndrome values of the primalgraph, whereas solid circles connected to dashed lines are syndromevalues of the dual graph. Note that the solid circles of FIGS. 17A-D areentirely different from the open circles shown in FIGS. 16A-E, whichdesignate qubits. As explained above in reference to FIGS. 11A-C,syndrome values may be calculated by performing a parity check once allof the syndrome graph edges connected to the syndrome value have beenmeasured. A single fusion measurement will measure both a single edge inthe primal graph and a single corresponding edge in the dual graph, twoexamples of which are shown in FIG. 17A. More generally, each primaledge and dual edge pair that intersects at their midpoint will have bothedges measured by a single fusion measurement.

FIG. 17B illustrates the same syndrome graphs as FIG. 17A, but with thelattices separated for clarity. The arrows illustrate the correspondencebetween two sets of primal/dual edges, where the two edges indicated byan arrow will be measured by a single fusion measurement.

FIG. 17C illustrates the same syndrome graphs as FIGS. 17A-17B after asubset of the sequence of fusion measurements have been performed. Asillustrated, bold solid black edges indicate that the fusion measurementresulted in an erasure, whereas non-bold solid edges indicate that thefusion measurement was successful. Unmeasured edges are indicated withdashed lines. For clarity, primal and dual edges are not distinguishedby line type in FIGS. 17C-D, rather, they are separated (left and right,respectively).

As used herein, the term “connected component” refers to any contiguousset of erased edges in either the primal or dual syndrome graph. Forexample, a connected component composed of seven erased edges is shownin the primal graph on the left side of FIG. 17C, whereas the dual graphshown on the right side of FIG. 17C has a larger number of smallerconnected components (e.g., 4 connected components of size two, one,one, and one). An edge may be erased by either a loss measurement resultor a failed measurement result. As used herein, any node that isconnected to an edge of a connected component is considered to be“within” the connected component.

In the example shown in FIG. 17C, the subsequent fusion measurement tobe performed involves the two edges indicated by the double-headed arrow(one primal edge and one dual edge). According to embodiments herein,the basis for performing this fusion measurement may be selected basedon the results of previous fusion measurements. The basis may beselected to reduce the likelihood of the entire sequence of fusionmeasurements resulting in a connected component that spans the lattice,thus reducing the likelihood of an overall logical error. A failureoutcome will result in one of the primal or dual edges involved in themeasurement being erased, and the basis for performing the fusionmeasurement may be selected to determine which of the involved primal ordual edge will be erased in the event of a failure outcome. In otherwords, the basis may be selected to determine which of the involvedprimal or dual edge will risk erasure in the event of a failure result.Of course, a successful outcome will successfully measure both of theinvolved primal and dual edges. While a loss outcome of a fusionmeasurement will erase both edges, a loss outcome is relatively lesscommon than either a success outcome or a failure outcome. Accordingly,as shown in the numerical examples given below in reference to FIGS.21-22, performing adaptive basis selection to selectively allocateerasures from failure results to either the primal or dual graph maysubstantially improve the fault tolerance of the FBQC code.

In the example shown in FIG. 17C, the two ends of the primal edgeinvolved in the fusion measurement are part of the same connectedcomponent. Accordingly, an erasure of this primal edge will not increasethe likelihood of an overall logical error, and it may be desirable toselect the basis for the fusion measurement such that this primal edgeis risked. Said another way, in some embodiments, in this case thefailure basis is selected such that the dual edge is successfullymeasured and primal edge is erased in the case of a failure result. Thisis an example of a “trivial” decision where an erasure of one edgeinvolved in the fusion measurement does not increase the likelihood of alattice-spanning connected component, so that the basis is selected torisk erasure for this edge.

FIG. 17D illustrates an alternative scenario, similar to FIG. 17C butwith a different set of prior fusion measurement results. Similar toFIG. 17C, bold solid black edges indicate that a fusion measurement ofthe edge resulted in an erasure, non-bold solid edges indicate that thefusion measurement was successful, and unmeasured edges are indicatedwith dashed lines. The two edges involved in the next fusion measurementare indicated with the double-headed arrow. However, in contrast to thetrivial case illustrated in FIG. 17C, a more subtle determinationutilizing an exposure comparison is employed for the graph illustratedin FIG. 17D to select the basis for the fusion measurement.

In the example illustrated in FIG. 17D, an erasure of either of theinvolved primal or dual edges will increase the likelihood of a logicalerror. To determine which edge to risk (i.e., which edge will be erasedin the event of a failure), an exposure may be calculated for each ofthe involved primal and dual edges.

The “exposure” of an edge is defined herein as follows. Each edgeconnects two nodes of either the primal or dual graph. Often, one orboth of these nodes may be part of one or more respective connectedcomponents. For example, the left node connected to the indicated edgeof the primal graph in FIG. 17D is part of a 6-node connected component,and the right node connected to the indicated edge of the primal graphis part of a different 6-node connected component. The exposure of anedge is calculated as the product of the exposures of these twoconnected components. The exposure of a connected component, in turn, isdefined as a sum (either weighted or unweighted) of the unmeasured edgesand/or the measured edges adjacent to the connected component, notcounting the edge for which the exposure is being calculated. Forexample, the left connected component adjacent to the indicated edge ofthe primal graph in FIG. 17D may be determined to have an exposure of 3,whereas the right connected component adjacent to the indicated edge ofthe primal graph in FIG. 17D has an exposure of 6, leading to an overallexposure of the indicated edge of the primal graph of 3*6=18.

In some cases, the edge may be adjacent to a node that is not part of aconnected component. For example, the indicated edge in the dual graphof FIG. 17D is directly above a connected component of size 3, butdirectly below a region of unmeasured nodes that does not contain aconnected component. For such nodes that are not part of a connectedcomponent, the exposure of the node is set equal to 3 to account for thethree unmeasured edges (not counting the indicated edge) adjacent tothis node. Accordingly, the indicated edge of the dual graph of FIG. 17Dhas an exposure of 3*1=3.

In FIG. 17D, since the indicated primal edge has a higher exposure, insome embodiments, the fusion failure basis is chosen such that theprimal edge is successfully measured and the dual edge is erased in thecase of a failure result.

FIG. 18 illustrates a 3D lattice with two connected components (i.e.,two contiguous clusters of erased edges) indicated with bold lines.

These examples are illustrative. The choice of error correcting codedetermines the set of qubit pairs that are fused from certain resourcestates, such that the output of the qubit fusion system is the classicaldata from which the syndrome lattice may be directly constructed. Insome embodiments, the classical error syndrome data is generateddirectly from the qubit fusion system without the need to preformadditional single particle measurements on any remaining qubits. In someembodiments, the joint measurements performed at the qubit fusion systemare destructive of the qubits upon which joint measurement is performed.

Section II. Adaptive Basis Selection for Fusion Measurements

Embodiments herein present systems and methods for performing adaptivebasis selection for fusion measurements for constructing entangledquantum states and executing quantum algorithms. Advantageously, theloss tolerance of Lattice Preparation Protocols (LPPs) may besignificantly increased by adaptively choosing the failure basis toobtain fusions that are more important for avoiding logical error. Asone example, methods described herein may increase the loss tolerance ofan LPP based on 6 qubit resource states (which includes placing failuresin 2D sheets) from 0.4% per photon in the non-adaptive case to 1.4% perphoton (including boosting measurements, as explained below).

Errors (e.g., erasures) in a fault-tolerance lattice cause logicalfailure if they link up in a way that spans the logical qubit.Embodiments herein employ fusion basis adaptivity to organize erasurescoming from fusions in a way that reduces the likelihood that they linkup to cause a logical error, enabling a significant increase in errortolerance. The reorganization of errors into a docile configuration(i.e., a configuration that does not result in logical error) may beachieved simply by adaptively changing the basis of the fusionmeasurement. Fusion adaptivity can be shown to be extremely effective inincreasing the loss threshold for fault tolerant quantum computing.

Three outcome results are possible in a type-II fusion measurement.First, a successful outcome of a fusion measurement results insuccessful entanglement of the resource states involved in the fusionmeasurement for both the primal and dual syndrome graphs. Second, afailure outcome results in a single qubit measurement wherein a fusionmay be obtained in one of the primal or dual syndrome graphs, while thesecond fusion of the other syndrome graph is lost. The fusion that issuccessfully obtained for a failure result may be chosen based onselective application of local Clifford gates before the Bellmeasurement. Said another way, the basis of a fusion measurement may beselected such that a failure result will cause an error in an edge ofeither the primal or the dual syndrome graph of the entangled state,whereas the measurement of the other edge of the primal or dual syndromegraph still succeeds. Third, a loss outcome of a fusion measurementresults when one or more photons escape during the measurement and arenot detected, resulting in a failed fusion for both the primal and dualsyndrome graphs.

Embodiments herein consider results of previous measurements beforeperforming a subsequent fusion measurement, use this information todetermine which edge is more important for avoiding a logical error(i.e., the edge in the primal or dual graph), and then choose thefailure basis of the fusion measurement such that this edge is obtainedeven when the fusion measurement fails. Advantageously, embodimentsherein may be used in any syndrome graph and may result in significantincrease in loss tolerance even with relatively small unencoded states.

Syndrome Graphs

The results of a type-II fusion measurement may be understood in termsof its effect on edges in a syndrome graph of the entangled state. Eachtype-II fusion measurement measures one edge in the primal graph and onein the dual graph of the syndrome graph. A success results in theerasure of neither edge, a loss results in the erasure of both edges,and a failure results in the erasure of either the primal or dual edge,depending on the basis of the fusion measurement. A chain of erasuresthat spans the logical qubit results in a logical error. Beforeperforming a fusion measurement, embodiments herein determine which edgeis more likely to result in a spanning chain of erasures after all edgesof the syndrome graph have been measured, and a fusion basis is chosenfor the measurement such that this edge is not erased even in the caseof a failure outcome.

Edges that exist in the syndrome graph may be classified as one of threetypes, 1) successful edges for which the parity has been obtainedsuccessfully, 2) failed edges for which the parity measurement wasattempted but was not be obtained due to a loss or failure (these edgesare erased in the syndrome graph), and 3) un-attempted edges which havenot yet been measured. Un-attempted edges may or may not be erased inthe future depending on the outcome of future fusion measurements.

Various metrics may be employed to determine which edge to risk in afusion measurement. As a first possibility, a trivial rule may beapplied whereby, if an edge is connecting two nodes that are part of thesame connected component, the erasure of that edge may not increase theprobability of a spanning path since its neighbors already have analternate path connecting them. For example, as shown schematically inFIG. 19, the illustrated edge connects two portions of a singleconnected component. Hence, whenever possible, such an edge may bepreferentially risked to protect the complementary edge in the other(e.g., primal or dual) syndrome graph.

In some embodiments, an un-attempted edge may be unable to be part of aloop of erasures, even in the future (i.e. if there is no other pathcomposed of failed and un-attempted edges connecting its adjacentnodes), and this edge may not be able to contribute to a logical error.Hence, this edge may be risked without increasing the probability of alogical error. Accordingly, in some embodiments, this type ofun-attempted edge may be selectively risked in a fusion measurement.

For the remaining edges (i.e., where the trivial rule may not apply),embodiments herein present decision-making strategies to determine whicherror configuration is better or worse given the outcome of previousmeasurements. In some embodiments, as described in greater detail below,a strategy referred to herein as Exposure Based Adaptivity (XBA) isutilized that results in high error tolerance in the regime of highloss, and allows for computationally efficient decision making.

Exposure Based Adaptivity (XBA)

A connected component in a syndrome graph is a set of contiguous erasededges. XBA attempts to prevent the growth of the more dangerous of theconnected components in either the primal or dual syndrome graphs (i.e.,the one that is more likely to lead to a logical error). If every edgewere either erased or successfully measured in an independent andidentically distributed (i.i.d.) random fashion in a large lattice, thepresence of either a spanning path of erased edges (i.e. logical error)or a connected component of the same size as the full lattice haveroughly the same probability of occurring. Although the size of theconnected components adjacent to an edge may seem to be a good heuristicfor determining the importance of an edge, this may not always be aneffective metric. For example, if most of the edges adjacent to a largecomponent are successful edges, the connected component has very limitedpotential for growth. For example, as shown in FIG. 20, even though thesize of the connected component is large, it is mainly surrounded bysuccessful edges. Hence, it's potential for growth, and of being part ofa future spanning path, is dictated by the smaller illustrated “regionof growth”. Such a situation where most of the edges adjacent to aconnected component are successful is quite likely in a linear opticalquantum computer, as one dimension of the lattice is time.

The XBA metric defines the exposure of a connected component as a sum ofun-attempted edges adjacent to the connected component. The sum may beweighted or unweighted as described in greater detail below, in variousembodiments. Advantageously, this metric provides an effective indicatorof a connected component's potential for growth. The rule for applyingthe XBA metric is to risk the edge of either the dual or primal syndromegraph for which the product of the exposures of the components at thetwo ends of the respective edge is smaller. The XBA metric may beutilized for cases where the end points of both edges connect toneighbor nodes that are not already part of the same connectedcomponent. In these cases, where the two nodes on either side of an edgeto be measured are already connected by a single connected component(i.e., when the edge is an intra-cluster edge), the trivial selectionrule described above may be employed (i.e., the basis will be selectedto risk this edge).

In some embodiments, the XBA metric may be generalized to include aweighting that considers the relative importance of measurements in theprimal and dual syndrome graphs. For example, consider a fusionmeasurement that performs a probabilistic projective measurement tomeasure a particular primal edge in the primal syndrome graph and aparticular dual edge in the dual syndrome graph. In some embodiments,the calculation of the exposures of the primal and dual edge may utilizea parameter β, where β=0 when the dual edge is an intra-cluster edge,β=1 when the primal edge is an intra-cluster edge and the dual edge isnot,

$\beta = {{\frac{1}{2}\left( \frac{\chi_{dual}}{\chi_{primal}} \right)^{s}\left( {1 - q} \right)} + {{0.5}q}}$when χ_(dual)≥χ_(primal) and neither edge is an intra-cluster edge, and

$\beta = {{\left\lbrack {1 - {\frac{1}{2}\left( \frac{\chi_{dual}}{\chi_{primal}} \right)^{s}}} \right\rbrack\left( {1 - q} \right)} + {{0.5}q}}$when χ_(dual)<χ_(primal) and neither edge is an intra-cluster edge. Hereχ_(primal) and χ_(dual) are the exposures of the primal edge and thedual edge, respectively, and q and s are numerically optimizedparameters. For example, a quantum computing test algorithm may beperformed a plurality of times for different combinations of values of qand s, and it may be determined which combination of values of thetested combinations produces the most desirable fault tolerance for thetest algorithm. The parameter β may be used to choose one or moreparameters and/or settings for the fusion measurement being performed.For example, in the case of encoded fusion there are many choices forperforming the fusion which may have different probabilities of primaland dual erasure. In such a scenario, β may be computed according to theequations above, and may be used to determine a minimum value of a costfunction ƒ:ƒ=(1−β)*Prob(primal erasure)+β*Prob(dual erasure)  (17)

In the case of unencoded fusions, the two choices of a failure basis maybe XX failure and ZZ failure. In some embodiments, the cost functionshown in Eq. (17) may be computed for each of the XX and ZZ failurebases (e.g., the probability of an erasure in the primal and dual graphsmay vary depending on the selected failure basis), and the failure basisthat results in a smaller cost function may be selected for performingthe fusion measurement. In other embodiments, the definition of moregeneral and may work for encoded fusions with other failure basischoices as well.

Numerical Results

FIG. 21 is a graph illustrating numerical results for loss rates perphoton for a variety of methodologies, according to various embodiments.In producing the data illustrated in FIG. 21, it was assumed that eachphoton in the resource state and each photon in the boosting resourcestate experiences the same probability of loss, which is referred to asthe loss per photon l. Type-II fusion boosted with a Bell pair wasutilized. Accordingly, the probability of a non-loss outcome isη=(1−l)⁴. The success probability is ¾η≈¾(1−4l), the failure probabilityis ¼η≈¾(1−4l), and the loss probability is 1−η≈4l.

In FIG. 21, the threshold value of loss per photon is computed bycalculating the wrapping probability of a spanning series of connectedfailed edges on lattices of size 25³ (dashed lines) and 50³ (solidlines). A fusion measurement process without selective basis adaptivity(purple) corresponds to the 6 ring LPP where failures in the absence ofloss form disconnected sheets. While such an LPP may tolerate 34.7%failures in the absence of loss, any loss starts connecting the sheetscaused by failure resulting in a relatively small loss tolerance perphoton of 0.4% (purple lines). For the same syndrome graph, using fusionadaptivity with XBA (red lines), a per photon loss tolerance of 1.4% isobtained.

The wrapping probability is used in FIG. 21 to estimate the photon lossthreshold because it is less sensitive to finite size effects than thespanning probability. It may be shown, however, that spanningprobability and size of the largest component exhibit similar resultsfor non-adaptive and XBA adaptive basis selection.

In the results illustrated in FIG. 21, the exposure of a measured edgeis defined to be zero. However, in a fusion measurement scenario with asignificant probability of Pauli errors, more measurements surrounding acluster may make it more likely that an error has occurred. Accordingly,in some embodiments, tolerance to Pauli errors may be improved byassigning a non-zero exposure to every unmeasured edge. This parameter,which is referred to herein as accretion, has a small impact on theloss-only results but becomes important once Pauli errors areintroduced.

The XBA rule used above gives each connected component involved in thefusion measurement a score based on the number of exposed (i.e.,unmeasured) edges, but the XBA rule does not consider how far apartthose exposed edges are. However, exposed edges that are far apart inthe connected component may be more likely to cause a graph-spanningconnected component, and may accordingly be more likely to result in alogical error than exposed edges that are nearby to each other. Toaccount for this, in some embodiments, the second moment of the exposededges of a connected component about their mean position may be used todetermine the overall exposure of a connected component. Additionally oralternatively, while the XBA rule described above only looks at theexposure of the immediate neighborhood of a connected component, inother embodiments the exposure of the extended neighborhood of aconnected component may be considered. For example, a connectedcomponent with a large exposure one un-attempted edge away from theconnected component may be considered to have a larger exposure, in someembodiments.

Adaptive Boosting

In some embodiments, in addition to adaptively choosing the basis forfusion measurements, an adaptive level of boosting may be chosen for thefusion measurement process. “Boosting” refers to the introduction ofredundancy by utilizing multiple sets of photons for performing a singlefusion measurement. For example, as described in greater detail in theAppendix below, boosting may utilize ancillary Bell pairs or pairs ofsingle photons to increase the probability of a success outcome.Boosting increases the probability of obtaining a successful outcome fora fusion measurement, while decreasing the probability of a failureoutcome. The inclusion of additional photons increases the probabilityof a loss outcome, but this may be more than compensated for by thedecrease in the probability of a failure outcome, at least in someembodiments. This section describes methods for adaptive boosting,whereby boosting is turned on or off based on the results of previousfusion measurements. Boosting may be turned off if one of the edges(i.e., from the primal or dual syndrome graph) measured in a fusionmeasurement does not increase the probability of a spanning path oferasures (e.g. where the edge connects two nodes that are already partof the same connected component). Boosting may be turned off in thesesituations, since only one of the edges is important and the basis maybe selected to guarantee a successful measurement of the important edgein the case of a failure outcome.

In some embodiments, if the product of the size of the connectedcomponents at either end of the edge corresponding to one of the primalor dual syndrome graphs is greater than that at the ends of the otheredge by a predetermined factor γ_(b), one of the edges may be expectedto be much more important than the other, and boosting may be turnedoff. The factor γ_(b) may be tuned numerically (e.g., empirically) toincrease the resultant photon loss threshold.

FIG. 22 is a graph comparing photon loss percentages using no basisselection adaptivity, basis selection adaptivity with boosting alwayspresent, and basis selection adaptivity with boosting selectivelyapplied. The data illustrated in FIG. 22 was derived for a Kagome-24lattice, and the solid and dashed line are simulations for lattices ofsize 48³ and 24³, respectively. The threshold is determined based on thecrossing of the solid and dashed lines. As illustrated, fusionadaptivity and adaptive boosting both increase the photon lossthreshold, and adaptive boosting improves the photon loss thresholdcompared to always boosting. Table 1 tabulates results for photon lossthresholds (LTs) for a variety of different lattice geometries.

TABLE 1 LT: Fusion LT: Fusion LT: No Adaptivity, Adaptivity, FusionGeometry Adaptivity Always Boost Adaptive Boost Kagome-6 0.4% 1.4% 1.5%Kagome-10   1% 2.2% 2.3% Kagome-16   2% 3.4% 3.8% Kagome-24 2.8% 4.2%4.8% Kagome-∞ 2-strands 2.5% 3.4% 3.8%

As shown in Table 1, schemes with larger resource states tend to havebetter loss thresholds. One interesting comparison is between Kagome-16and Kagome-∞ 2-strands. In the case of no adaptivity, the Kagome-∞2-strands lattice has a higher threshold since it uses infinitely longresource states which results in 1D failure subgraphs. However, whenfusion adaptivity is used (both with and without adaptive boosting), theloss threshold for Kagome-16 is the same as Kagome-∞ 2-strands, althoughKagome-16 uses much smaller resource states than Kagome-∞ 2-strands (16vs ∞). This may be because both schemes have the same degreedistribution in the full syndrome graph, and it appears that this is agood metric for evaluating the performance of fusion geometries forfusion adaptivity with the XBA rule.

The no adaptivity cases and the always boost cases here use singleboosting with Bell pairs. In some cases, it is possible to run theseschemes un-boosted e.g. Kagome-16 with fusion adaptivity and Kagome-24,Kagome∞ 2-strands both with and without fusion adaptivity. However, forthe cases studied here, this results in a lower loss tolerance.

Advantageously, embodiments herein for adaptive basis selection and/oradaptive boosting provide significantly more loss tolerance withoutadditional active components in the path of a photon to incorporate thelocal Cliffords before the fusion measurements in the multiplexingnetwork.

Adaptive Basis Selection for Encoded Fusion

In some embodiments, a FBQC system may utilize encoded qubits forperforming fusion measurements in a quantum computing algorithm. Thesemethods may be referred to as “encoded fusion”, where each qubit isreplaced by multiple qubits that are entangled together, to collectivelyform a single encoded qubit. In these embodiments, the term “physicalqubit” is used to refer to the individual qubits that collectively forman encoded qubit. Each physical qubit may be a dual-rail photonic qubit,as one example.

For example, a typical (non-encoded) fusion measurement is schematicallyillustrated in FIG. 23, where a fusion measurement is to be performed onqubits 1 and 2. In FIG. 23, qubit 1 is entangled with qubits 3 and 4,whereas qubit 2 is entangled with qubits 5 and 6. The fusion measurementon qubits 1 and 2 may perform a projective measurement on thestabilizers X₁X₂ and Z₁Z₂. Said another way, the fusion measurement mayproject the state of the qubits 1 and 2 onto the stabilizers X₁X₂ andZ₁Z₂. In linear optical quantum computing, fusion measurements areprobabilistic with a particular likelihood of success, failure, andloss, as described in greater detail above. If the fusion successfullyperforms both of these stabilizer measurements, the qubits 3, 4, 5, 6will be entangled in a desired way. For example, measurement outcomesfor success and failure results for various specific examples of fusionmeasurements are illustrated in FIG. 34. While FIG. 23 illustrates oneconfiguration of external qubits (3, 4, 5, 6), more generally there maybe any number of qubits pre-entangled with qubits 1 and 2.

Fusion measurements in linear optics don't measure X₁X₂ and Z₁Z₂deterministically. For example, even in the absence of loss (i.e.,photon escape, absorption, etc.), a failure outcome may occur with aprobability of 50%, where one of X₁X₂ or Z₁Z₂ is not successfullymeasured (e.g., either X₁X₂ or Z₁Z₂ will be unsuccessfully measureddepending on the selected basis of the fusion measurement).

To increase the probability of successfully obtaining both measurements,encoded fusion may be employed. In encoded fusion, the two qubits to befused may be replaced by composite qubits, i.e., encoded qubits that areeach made from a collection of two or more entangled physical qubits.The encoded qubits may be encoded using a small error correcting code,e.g., using a (2,2) Shor code, or any other quantum error correctingcode. For example, qubits 1 and 2 may be encoded and are therebyreplaced with multiple entangled qubits, e.g., 4 qubits each as shown inFIG. 24A. In FIG. 24A, “1_1” denotes the first qubit within the encodedqubit 1, “1_2” denotes the second qubit within the encoded qubit 1,“2_1” denotes the first qubit within encoded qubit 2, etc. Asillustrated, the dashed box represents an encoded qubit (i.e. the qubitsinside the dashed boxed are pre-entangled in some way, e.g., accordingto a small error correcting code). Similar to regular unencoded fusionmeasurements, encoded qubits may undergo an entangling measurementreferred to herein as an encoded fusion measurement. In someembodiments, an encoded fusion measurement may be a transversaloperation whereby the encoded fusion measurement includes a plurality ofapplications of individual fusion measurements on physical qubits of theencoded qubits, where the individual fusion measurements are of the sametype as the encoded fusion measurement. For example, in a transversaloperation, an encoded fusion measurement of X₁X₂ may be obtained, wherethe overbar indicates a value of the encoded qubits 1 and 2, byperforming X₁X₂ fusion measurements on individual physical qubit pairsof the encoded qubits.

In the examples that follow, the terminology of the stabilizer formalismfor quantum error correction is used. In the stabilizer formalism,encoded qubit states are used that take form of what are referred to asstabilizer states. Stabilizer states are given this name because theyare the +1/−1 eigenstates of a set of operators, called the stabilizeroperators (or simply the stabilizers). In some embodiments, thestabilizers are the set of n-qubit Pauli operators which leave the codeinvariant (i.e., the stabilizer states are invariant under operation byany of the stabilizers, except for an overall factor of +1 or −1).Stated another way, in the language of quantum measurements, thestabilizers that define the stabilizer code are the set of measurementsthat will return +1 eigenvalues when the state being measured (i.e., theencoded qubit) has no errors on the underlying physical qubits and willreturn one or more −1 eigenvalues when the state being measured has adetectable error on one or more of the underlying physical qubits. Inaddition, stabilizer measurements may be understood to be joint paritymeasurements, and (unlike individual qubit measurements) these paritymeasurements do not measure the quantum state of the encoded qubit, butrather only its parity. Accordingly, measurement of the stabilizers doesnot collapse the encoded qubit state, thereby leaving the underlyingquantum information intact.

In the example illustrated in FIG. 24A, the stabilizers for the qubitsinside the dashed boxes are Z_(i_1)Z_(i_2), Z_(i_3)Z_(i_4),X_(i_1)X_(i_2), and X_(i_3)X_(i_4) for both i=1 and i=2. In terms ofgraph states, the entangled qubits may be represented as shown in FIG.24B, where “H” denotes that the particular qubit has a Hadamard gateapplied to it. The physical fusion measurements may be performed in theorder shown by the arrows in FIGS. 24A and 24B.

The encoded fusion measurement shown in FIG. 24A performs a projectivemeasurement, and similar to the case of unencoded fusion, results in aprobabilistic measurement of X₁X₂ and Z₁Z₂ , where the overline denotesthat the quantity refers to an overall value of the encoded qubit, andthe subscript under the overline is the label for the encoded operator.However, because of the encoding and the fact that there are stabilizersthat initially exist within an encoded qubit, there is more than one wayto measure the encoded operators. For example, for the illustratedexample,X ₁ X ₂ =X _(1_1) X _(2_1) X _(1_2) X _(2_2) =X _(1_3) X _(2_3) X _(1_4)X _(2_4)  (17)Z ₁ Z ₂ =Z _(1_1) Z _(2_1) Z _(1_3) Z _(2_3) =Z _(1_1) Z _(2_1) Z _(1_4)Z _(2_4) =Z _(1_2) Z _(2_2) Z _(1_3) Z _(2_3) =Z _(1_2) Z _(2_2) Z_(1_4) Z _(2_4)  (18)

These variations may be obtained by multiplying the encoded operator bythe stabilizers of the code. While X₁X₂ and Z₁Z₂ are the desired encodedfusion measurement outcomes, as used herein,X_(1_1)X_(2_1)X_(1_2)X_(2_2), X_(1_3)X_(2_3)X_(1_4)X_(2_4), etc. aresets of “effective measurement results” that may equivalently be used toobtain the desired encoded fusion measurement outcomes. The set ofeffective measurement results may be referred to as an effectivemeasurement group of the encoded fusion measurement. Since there aremultiple ways of obtaining a valid set of effective measurement resultsto measure the desired logical operators, the encoded measurements maybe obtained even when some of the measurements are erased.

For example, during the sequence of four fusion measurements shown inFIG. 24A (e.g., the top measurement is performed first, followed by thenext measurement down, etc., as indicated by the arrows), at eachsubsequent fusion measurement, depending on the outcome of the previousmeasurements, it may be that a failure of a particular X or Zmeasurement does not compromise the ability of the set of encoded fusionmeasurements to successfully measure both X₁X₂ and Z₁Z₂ . As oneexample, if X_(1_1), X_(2_1), X_(1_2) and X_(2_2) have all beenpreviously successfully measured, X₁X₂ may be determined according toequation (17) even if one or more of X_(1_3), X_(2_3), X_(1_4) andX_(2_4) are not obtained (e.g., from a failure result).

Embodiments herein present methods and devices whereby previousmeasurement outcomes within the sequence of encoded fusion measurementsare used to choose a basis for how a subsequent fusion measurement willbe performed. Accordingly, in the example above, it may be desirable toset the failure basis for the (1_3, 2_3) and (1_4, 2_4) fusionmeasurements to be XX, since a failure in the XX basis will still obtainthe Z measurements in a failure outcome, and a failure in the XX basiswill not compromise the overall encoded fusion measurement.

The examples here describe embodiments where each encoded fusionmeasurement is selected to be performed in either the XX or the ZZbasis. However, it may be appreciated that any two orthogonal bases maybe used for the encoded fusion measurements. In general, a basis forperforming each encoded fusion measurement is selected from twopotential orthogonal bases, where XX and ZZ are a typical set oforthogonal bases, in some embodiments.

More generally, in various embodiments the set of encoded fusionmeasurements may involve encoded qubits that are each encoded withdifferent numbers of qubits, and/or the encoded fusion measurements mayinvolve more than two encoded qubits (e.g., there may be a 3-way encodedfusion measurement, as described in greater detail below). In theseembodiments, the stabilizers and/or the effective measurement resultsthat correspond to the desired fusion measurement outcomes may vary.Accordingly, the specific method whereby a basis is selected for asubsequent fusion measurement may also vary. However, in each of theseembodiments, the basis may be selected from two orthogonal bases suchthat a failure result in the selected basis either a) does not preventthe measurement of the desired encoded fusion measurement outcomes(i.e., a particular basis may be selected if the encoded fusionmeasurement may be obtained even if a failure outcome occurs in theselected basis) or b) has a lower likelihood of preventing themeasurement of the desired encoded fusion measurement outcomes comparedto the alternative basis.

In some embodiments, each encoded fusion measurement may be performed inone of four different ways. It may be performed in one of two failurebases (e.g., X or Z) and may be performed with or without boosting. Aphoton may be lost in each fusion measurement with probability l, andη=1−l. The photon loss probability may be determined experimentally fora particular photonic system. In these embodiments, if a fusionmeasurement is un-boosted, it will succeed with probability ½ if nophoton is lost during the measurement. In this case, the probability ofsuccess is η²/2. The probability of failure is also η²/2. If eitherinput photon is lost, no measurement is obtained.

The fusion measurement may be boosted using a Bell pair to increase thesuccess probability to ¾ in the absence of photon loss. However, nowthere are now 4 photons going into a fusion, which increases theprobability of loss. It may be assumed that all of the photons(including the photons from the Bell pair) have the same lossprobability l. Further, by placing a Hadamard before the boosted qubit,the failure basis may be selected to be either X or Z, whereby themeasurement erased in case of failure may be chosen in both the boostedand un-boosted cases.

In these embodiments, the probabilities of each possible outcome foreach type of measurement are as follows.

For an un-boosted measurement where a failure successfully measures ZZ(and not XX), the probability of measuring both XX and ZZ is η²/2, theprobability of measuring ZZ but not XX is η²/2, and the probability ofnot measuring either XX or ZZ (i.e., for photon loss) is 1−η².

For an un-boosted measurement where a failure successfully measures XX(and not ZZ), the probability of measuring both XX and ZZ is η²/2, theprobability of measuring XX but not ZZ is η²/2, and the probability ofnot measuring either XX or ZZ (i.e., for photon loss) is 1−η².

For a boosted measurement where a failure successfully measures ZZ (andnot XX), the probability of measuring both XX and ZZ is 3η⁴/4, theprobability of measuring ZZ but not XX is η⁴/4, and the probability ofnot measuring either XX or ZZ (i.e., for photon loss) is 1−η⁴.

For a boosted measurement where a failure successfully measures XX (andnot XZZX), the probability of measuring both XX and ZZ is 3η⁴/4, theprobability of measuring XX but not ZZ is η⁴/4, and the probability ofnot measuring either XX or ZZ (i.e., for photon loss) is 1−η⁴.

Embodiments herein may determine whether to boost a measurement or notto increase the likelihood of a successful outcome, depending on thespecific value of l (and/or η). The outcomes described here may beinferred from detector clicks after a physical fusion is performed. Thefour options for performing the fusion measurement may be selected usinga switching network, as shown in FIG. 25. If required, the circuits forall four kinds of fusion measurements may be obtained using beamsplitters and passive phase shifters. Local encoding may be utilized todetermine which of these four measurement options to choose for a fusionmeasurement, depending on the outcomes that have been observed in thepast, to have the best probability of obtaining X₁X₂ and Z₁Z₂ for theencoded qubits.

FIG. 26—Flowchart for Performing Adaptive Encoded Fusion

FIG. 26 is a flowchart diagram illustrating a method for performingadaptive basis selection during a sequence of encoded fusionmeasurements. The method shown in FIG. 26 may be used in conjunctionwith any of the computer systems or devices shown in the above Figures,among other devices. For example, the method shown in FIG. 26 may beperformed by a photonic quantum computing device or system 1001 and/or1201 as illustrated in FIGS. 10 and 12, respectively. Utilization of thedescribed method may utilize one or more sets of waveguides, one or moresets of beam splitters that couple the respective waveguides to producea photonic state comprising a plurality of photonic qubits within theone or more sets of waveguides. The quantum computing system may furtherinclude a fusion controller (e.g., the fusion controller 1319 and/or1419 illustrated in FIGS. 13-15) configured to direct the describedmethod steps, and may be include (or be coupled to) a classicalcomputing system 1003/1207 for processing classic information anddirecting operations of the quantum computing device. In otherembodiments, the quantum computing system may utilize a general type ofmode structure, rather than optical waveguides, and may further utilizemore general forms of mode coupling (i.e., rather than beam splitters).Alternatively, in some embodiments the methods described in FIG. 26 maybe utilized in a quantum communication network, quantum internet, ormore generally in any application where it is desired to perform aprojective encoded fusion measurement on two or more encoded qubits. Itis to be understood this method may be used by any of a variety of typesof photonic quantum computing architectures, and these other types ofsystems should be considered within the scope of the embodimentsdescribed herein.

FIGS. 24A-B and FIG. 27 illustrate additional supporting material tohelp explain the methods described in association with FIG. 26. Inparticular, FIGS. 24A-B illustrate an example set of first and secondencoded qubits that may serve as inputs to an encoded fusionmeasurement, and FIG. 27 is a system diagram illustrating a qubit fusionsystem 1305 that may be used to implement the methods described inassociation with FIG. 26. Reference numbers from these Figures areutilized to clarify potential relationships between the describedmethods and the Figures, in some embodiments. The system shown in FIG.27 shares some components with the system shown in FIG. 15, but differsin that FIG. 27 explicitly illustrates first and second fusion sites.

In various embodiments, some of the elements of the scheme shown may beperformed concurrently, in a different order than shown, or may beomitted. Additional and/or alternative elements may also be performed asdesired. As shown, the method of FIG. 26 may operate as follows.

At 2602, a first fusion measurement 2414 is performed on a firstphysical qubit 2401 of a first encoded qubit 2410 and a second physicalqubit 2405 of a second encoded qubit 2412. The first fusion measurementmay be performed at a first fusion site 1501 and may be directed by thefusion controller 1319.

The first encoded qubit may include a first plurality of physical qubitsincluding the first physical qubit, and the second encoded qubit mayinclude a second plurality of physical qubits including the secondphysical qubit. The first encoded qubit and the second encoded qubit maybe received from a source or multiple sources. The first fusionmeasurement may be a Type-II fusion measurement, or a Type-I fusionmeasurement. A first classical measurement result 1540 of the firstfusion measurement may be obtained. The first classical measurementresult may be obtained at a first fusion site 1501 and transmitted tothe fusion controller 1319. The first classical measurement result maybe stored in a non-transitory memory medium.

At 2604, a basis (1541, 1542) is selected for performing a second fusionmeasurement 2416 based at least in part on the first classicalmeasurement result 1540. The second fusion measurement may be performedat a second fusion site 1546 and may be directed by the fusioncontroller 1319. The basis may be selected from a set of two orthogonalmeasurement bases. The selected basis may determine which of twomeasurement outcomes will fail in the case of a failure outcome of thefusion measurement. For example, when the failure basis is selected asX₁X₂ and a failure occurs, X₁X₂ may not be measured while Z₁Z₂ issuccessfully measured. Note that the subscripts correspond to the firstand second physical qubits. Some examples of possible outcomes forseveral specific fusion measurements for both success and failure areillustrated in FIG. 34.

In some embodiments, selecting the basis for performing the secondfusion measurement based at least in part on the first classicalmeasurement result includes determining that, when the second fusionmeasurement fails in a first basis, the encoded qubit measurement resultmay still be determined based at least in part on the first and thesecond classical measurement results. In these embodiments, the firstbasis is selected as the basis for performing the second fusionmeasurement based at least in part on determining that, when the secondfusion measurement fails in the first basis, the encoded qubitmeasurement result may still be determined based at least in part on thefirst and second classical measurement results.

In some embodiments, selecting the basis for performing the secondfusion measurement includes determining, based on stabilizers of thefirst and second encoded qubits, a plurality of effective measurementresults that correspond to the encoded fusion measurement result.Because of the redundancy introduced by the plurality of physical qubitsof an encoded qubit, an encoded fusion measurement results may beobtained through one or multiple effective measurement results, whereeach effective measurement result is a combination of measurements ofthe individual physical qubits that is equivalent to the encoded fusionmeasurement result. For example, Equations 17 and 18 above show oneexample of the set of effective measurement results that correspond toeach of two encoded fusion measurement results. In these embodiments,the selected basis may be selected such that a first effectivemeasurement result of the plurality of effective measurement resultsincludes the first classical measurement result and the second classicalmeasurement result when the second fusion measurement fails in theselected basis. In other words, it may be determined that when thesecond fusion measurement fails in a particular basis, an effectivemeasurement result that corresponds to the desired encoded fusionmeasurement result may still be determined based on the measurement thatis still obtained in the case of a failure outcome.

In some embodiments, it is determined, based at least in part on thefirst classical measurement result, whether to employ boosting for thesecond fusion measurement. Determining whether to employ boosting mayinclude determining whether the encoded fusion measurement result isdeterminable when the second fusion measurement fails in the selectedbasis, wherein it is determined not to employ boosting when the encodedfusion measurement result is determinable when the second fusionmeasurement fails in the selected basis. Alternatively or additionally,in some embodiments determining whether to employ boosting includesdetermining a photon loss rate of the second fusion measurement,determining loss probabilities for performing the second fusionmeasurement with and without boosting based on the photon loss rate,determining success probabilities for performing the second fusionmeasurement with and without boosting based on the loss probabilitiesand based on failure probabilities for performing the second fusionmeasurement with and without boosting, and determining whether to employboosting based on a comparison of the success probabilities forperforming the second fusion measurement with and without boosting.

At 2606, the second fusion measurement (1546, 2416) is performed on athird physical qubit 2402 of the first encoded qubit 2410 and a fourthphysical qubit 2406 of the second encoded qubit 2412 according to theselected basis. A second classical measurement result 1544 of the secondfusion measurement is obtained. The second classical measurement resultmay be obtained at a second fusion site 1546 and transmitted to thefusion controller 1319. The second classical measurement result may bestored in a non-transitory memory medium.

At 2608, an encoded fusion measurement result is determined of a fusionof the first encoded qubit 2410 and the second encoded qubit 2412 basedat least in part on the first 1540 and second 1544 classical measurementresults.

In some embodiments, one or more bases are sequentially selected forperforming one or more subsequent fusion measurements on additionalphysical qubits from the first and second plurality of physical qubits,wherein the one or more bases are selected based at least in part onclassical measurement results of previous fusion measurements. Forexample, each subsequent fusion measurement may be performed in a basisthat is selected based on the results of one or more previousmeasurements. While the method steps of FIG. 26 describe only first andsecond fusion measurements, this represents the simplest case andtypically an encoded fusion measurement result is obtained from encodedqubits that each include more than two qubits (e.g., 4, 6, 24, etc.).The one or more subsequent fusion measurements may be performed,obtaining one or more subsequent classical measurement results In theseembodiments, determining the encoded fusion measurement result isperformed further based at least in part on the one or more subsequentclassical measurement results.

At 2610, the encoded fusion measurement result is stored in anon-transitory memory medium. The encoded fusion measurement result maybe used in a fusion based quantum computing (FBQC) algorithm. Forexample, an output of a FBQC algorithm may be computed based at least inpart on the encoded fusion measurement result, and the output may bestored in the non-transitory memory medium.

FIGS. 28-29—Encoded 3-Way Fusion

FIG. 28 is a schematic illustration of a 3-way encoded fusionmeasurement, according to some embodiments. In a 3-way fusion operation,it may be desired to measure the operators X₁X₂X₃ , Z₁Z₂ and Z₂Z₃ . Insome embodiments, methods described herein for adaptive basis selectionmay be utilized in a 3-way encoded fusion measurement.

FIG. 29 is a flowchart diagram illustrating a method for performingadaptive basis selection during a 3-way encoded fusion measurement suchas that illustrated in FIG. 28. The method shown in FIG. 29 anddescribed below may incorporate one or more implementation details asdescribed above in reference to FIG. 26, as appropriate. The methodshown in FIG. 29 may be used in conjunction with any of the computersystems or devices shown in the above Figures, among other devices. Forexample, the method shown in FIG. 29 may be performed by a photonicquantum computing device or system 1001 and/or 1201 as illustrated inFIGS. 10 and 12, respectively. Utilization of the described method mayutilize one or more sets of waveguides, one or more sets of beamsplitters that couple the respective waveguides to produce a photonicstate comprising a plurality of photonic qubits within the one or moresets of waveguides. The quantum computing system may further include afusion controller (e.g., the fusion controller 1319 and/or 1419illustrated in FIGS. 13-15) configured to direct the described methodsteps, and may be include (or be coupled to) a classical computingsystem 1003/1207 for processing classic information and directingoperations of the quantum computing device. In other embodiments, thequantum computing system may utilize a general type of mode structure,rather than optical waveguides, and may further utilize more generalforms of mode coupling (i.e., rather than beam splitters). It is to beunderstood this method may be used by any of a variety of types ofphotonic quantum computing architectures, and these other types ofsystems should be considered within the scope of the embodimentsdescribed herein.

In various embodiments, some of the elements of the scheme shown may beperformed concurrently, in a different order than shown, or may beomitted. Additional and/or alternative elements may also be performed asdesired. As shown, the method of FIG. 29 may operate as follows.

At 2902, a first fusion measurement is performed on a first physicalqubit of a first encoded qubit and a second physical qubit of a secondencoded qubit. A first classical measurement result is obtained of thefirst fusion measurement.

At 2904, a first basis is selected for performing a second fusionmeasurement based at least in part on the first classical measurementresult.

At 2906, the second fusion measurement is performed on a third physicalqubit of the second encoded qubit and a fourth physical qubit of a thirdencoded qubit according to the first basis. A second classicalmeasurement result of the second fusion measurement is obtained.

At 2908, a second basis is selected for performing a third fusionmeasurement based at least in part on the first and second classicalmeasurement results.

At 2910, the third fusion measurement is performed on a fifth physicalqubit of the third encoded qubit and a sixth physical qubit of the firstencoded qubit according to the second basis. A third classicalmeasurement result of the third fusion measurement is obtained.

At 2912, an encoded fusion measurement result is determined of a fusionof the first, second and third encoded qubits based at least in part onthe first, second and third classical measurement results.

At 2914, the encoded fusion measurement result is stored in anon-transitory memory medium.

APPENDIX—EXAMPLES OF FUSION GATES

The following paragraphs describe additional detail and examples ofimplementations of fusion gates (and/or fusion circuits) for photonicqubits that may be used according to some embodiments that utilize TypeII fusion measurements. For example, the gates and circuits illustratedherein may be utilized to implement adaptive basis selection to generatean error-corrected logical qubit for fusion-based quantum computing, insome embodiments. It should be understood that these examples areillustrative and not limiting.

A Type II fusion circuit (or gate), in the polarization encoding, maytake two input modes, mix them at a polarization beam splitter (PBS),and then rotate each of them by 45° before measuring them in thecomputational basis. FIG. 30 shows an example. In the path encoding, aType II fusion circuit takes four modes, swaps the second and fourth,applies a 50:50 beam-splitter between the two pairs of adjacent modesand then detects them all. FIG. 31 shows an example.

Fusion gates may be used in the construction of larger entangled statesby making use of the so-called “redundant encoding” of qubits. This mayconsist in a single qubit being represented by multiple photons, i.e.:α|0

+β|1

→α|0

^(⊗n)+β|0

^(⊗n),

so that the logical qubit is encoded in n individual qubits. This may beachieved by measuring adjacent qubits in the X basis.

This encoding, denoted graphically as n qubits with no edges betweenthem, has the advantage that a Pauli measurement on the redundant qubitsdoes not split the cluster, but rather removes the photon measured fromthe redundant encoding and combines the adjacent qubits into one singlequbit that inherits the bonds of the input qubits (potentially adding aphase). In addition, another advantage of this type of fusion is that itis loss tolerant. Both modes are measured, so there is no way to obtainthe detection patterns that herald success if one of the photons islost. Finally, Type II fusion does not require the discriminationbetween different photon numbers, as two detectors need to click for theheralding of successful fusion and this can only happen if the photoncount at each detector is 1.

The fusion succeeds with probability 50%, when a single photon isdetected at each detector in the polarization encoding. In this case, iteffectively performs a Bell state measurement on the qubits that aresent through it, projecting the pair of logical qubits into a maximallyentangled state. When the gate fails (as heralded by zero or two photonsat one of the detectors), it performs a measurement in the computationalbasis on each of the photons, removing them from the redundant encoding,but not destroying the logical qubit. The effect of the fusion in thegeneration of the cluster is depicted in FIGS. 32A-D, where (A) showsthe measurement of a qubit in the linear cluster in the X basis to joinit with its neighbor into a single logical qubit at (B), and (C) and (D)show the effect that success and failure of the gate have on thestructure of the cluster. It can be seen that a successful fusion allowsbuilding of two-dimensional clusters.

A correspondence may be retrieved between the detection patterns and theKraus operators implemented by the gate on the state. In this case,since both qubits are detected, these are the projectors:

${h_{1}h_{2}},\left. {v_{1}v_{2}}\rightarrow\frac{{h_{1}h_{2}} + {v_{1}\nu_{2}}}{\sqrt{2}} \right.$${h_{1}v_{2}},\left. {v_{1}h_{2}}\rightarrow\frac{{h_{1}h_{2}} - {v_{1}v_{2}}}{\sqrt{2}} \right.$h₁², v₁² → ±h₁v₂ h₂², v₂² → ±v₁h₂,

where the first two lines correspond to ‘success’ outcomes, projectingthe two qubits into a Bell state, and the bottom two to ‘failure’outcomes, in which case the two qubits are projected into a productstate. A third outcome, ‘loss’, would result if either qubit escapes.

In some embodiments, the success probability of Type II fusion can beincreased by utilizing a process called “boosting”, whereby ancillaryBell pairs or pairs of single photons are used. Employing a singleancilla Bell pair or two pairs of single photons may boost the successprobability to 75%. As described in greater detail above, adaptiveboosting may be employed to adaptively determine when to apply boostingfor a fusion measurement, based on results of previous fusionmeasurements.

One technique used to boost the fusion gate comes from the realizationthat, when the fusion gate succeeds, it is equivalent to a Bell statemeasurement on the input qubits. Therefore, increasing the successprobability of the fusion gate corresponds to increasing that of theBell state measurement it implements. Two different techniques toimprove the probability of discriminating Bell states have beendeveloped by Grice (using a Bell pair) and Ewert & van Loock (usingsingle photons).

The former showed that an ancillary Bell pair allows achieving a successprobability of 75%, and the procedure may be iterated, usingincreasingly complex interferometers and larger entangled states, toreach arbitrary success probability (in theory). However, the complexityof the circuit and the size of the entangled states necessary may makethis impractical, in some implementations.

The second technique makes use of four single photons, input in twomodes in pairs with opposite polarization, to boost the probability ofsuccess to 75%. It has also been shown numerically that the proceduremay be iterated a second time to obtain a probability of 78.125%, but ithas not been shown whether this scheme is iterable to increase thesuccess rate arbitrarily.

FIG. 33 shows the Type II fusion gate boosted once using these twotechniques, both in polarization and path encoding. The successprobability of both circuits is 75%. The detection patterns that heraldsuccess of the fusion measurement are described below for the two typesof circuit.

When a Bell state is used to boost the fusion, the logic behind the‘success’ detection patterns is best understood by considering thedetectors in two pairs: the group corresponding to the input photonmodes (modes 1 and 2 in polarization and the top 4 modes inpath-encoding) and that corresponding to the Bell pair input modes(modes 3 and 4 in polarization and the bottom 4 modes in path-encoding).These may be referred to as the ‘main’ and ‘ancilla’ pairs,respectively. In these embodiments, a successful fusion is heraldedwhenever: (a) 4 photons are detected in total; and (b) fewer than 4photons are detected in each group of detectors.

When 4 single photons are used as ancillary resources, success of thegate is heralded whenever: (a) 6 photons are detected overall; and (b)fewer than 4 photons are detected at each detector.

When the gates succeed, the two input qubits are projected onto one ofthe four Bell pairs, as these may be all discriminated from each otherthanks to the use of the ancillary resources. The specific projectiondepends on the detection pattern obtained, as before.

Both of the boosted Type II fusion circuits, designed to take one Bellpair and four single photons as ancillae, respectively, may be used toperform Type II fusion with variable success probabilities if theancillae are not present or if only some of them are present (e.g., inthe case of the four single photon ancillae). This may be useful becauseit allows the employment of the same circuits to perform fusion in aflexible way, depending on the resources available. If the ancillae arepresent, they may be input in the gates to boost the probability ofsuccess of the fusion. If they are not, the gates may still be used toperform fusion with a lower but non-zero success probability.

As far as the fusion gate boosted using one Bell pair is concerned, acase to be considered is that of the ancilla being absent. In this case,the logic of the detection patterns heralding success may be understoodby considering the detectors in the pairs described above again. Thefusion is still successful when: (a) 2 photons are detected at differentdetectors; and (b) 1 photon is detected in the ‘principal’ pair and 1photon is detected in the ‘ancilla’ pair of detectors.

In the case of the circuit boosted using four single photons, multiplemodifications may be possible, removing all or part of the ancillae.This is analogous to the boosted Bell State Generator (BSG), which isbased on the same principle.

First consider the case of no ancillae being present at all. Asexpected, the fusion is successful with probability 50%, which is thesuccess rate of the non-boosted fusion. In this case, the fusion issuccessful whenever 2 photons are detected at any two distinctdetectors.

As for the boosted BSG, the presence of an odd number of ancillae turnsout to be detrimental to the success probability of the gate: if 1photon is present, the gate only succeeds 32.5% of the time, whereas if3 photons are present, the success probability is 50%, like thenon-boosted case.

If only two of the four ancillae are present, two effects are possible.If they are input in different modes in the polarization encoding, i.e.different adjacent pairs of ancillary modes in the path encoding, theprobability of success is lowered to 25%. However, if the two ancillaeare input in the same polarization mode, i.e. in the same pair ofadjacent modes in the path encoding, the success probability is boostedup to 62.5%. In this case, the patterns that herald success may beunderstood again by grouping the detectors in two pairs: the pair in thebranch of the circuit where the ancillae are input (group 1) and thepair in the other branch (group 2). This distinction is particularlyclear in the polarization-encoded diagram. Considering these groups, thefusion if successful when: (a) 4 photons are detected overall; (b) fewerthan 4 photons are detected at each detector in group 1; and (c) fewerthan 2 photons are detected at each detector in group 2.

In these examples, the fusion gates work by projecting the input qubitsinto a maximally entangled state when successful. The basis the state isencoded in may be changed by introducing local rotations of the inputqubits before they enter the gate, i.e. before they are mixed at the PBSin the polarization encoding. Changing the polarization rotation of thephotons before they interfere at the PBS yields different subspaces ontowhich the state of the photons is projected, resulting in differentfusion operations on the cluster states. In the path encoding, thiscorresponds to applying local beam-splitters or combinations ofbeam-splitters and phase shifts corresponding to the desired rotationbetween the pairs of modes that constitute a qubit (e.g., neighboringpairs in FIGS. 31 and 33). This may be useful to implement differenttypes of cluster operations, both in the success and the failure cases,which may improve efficiency of construction of a large cluster statefrom small entangled states.

FIG. 34 is a table illustrating the effects of several rotatedvariations of the Type II fusion gate used to fuse two small entangledstates. The diagram of the gate in the polarization encoding, theeffective projection performed, and the final effect on the clusterstate are shown.

Rotation to different basis states is further illustrated in FIG. 35,which shows examples of Type II fusion gate implementations for a pathencoding. Shown are fusion gates for ZX fusion, XX fusion, ZZ fusion,and XZ fusion. In each instance a combination of beam splitters andphase shifters (e.g., as described above) may be used, according tovarious embodiments.

Encoded 6-Ring Entangled Resource State

The following paragraphs describe a specific example of a 6-ring graphstate that may be implemented as encoded qubits, according to someembodiments. FIG. 36A is an illustration of the 6-ring graph state,which includes 6 numbered qubits. It may be shown that the illustrated6-ring graph state has six stabilizer measurements, which are X₁Z₂Z₆,Z₁X₂Z₃, Z₂X₃Z₄, Z₃X₄Z₅, Z₄X₅Z₆, and Z₅X₆Z₁. In general, a graph statewith nodes i and j has stabilizers that may be derived asX_(i)Π_(j∈∂i)Z_(j), where ∂i is the set of nodes adjacent to i on thegraph. In practice, it may be desirable to implement a resource statethat is more resilient to photon loss and fusion failure than the 6-ringillustrated in FIG. 36A. In some embodiments, each of the 6 qubits ofFIG. 36A may be encoded by a set of 4 physical qubits in an errorcorrecting code, resulting in a 24 encoded qubit resource state as shownin FIG. 36B. As illustrated, each of the 6 encoded qubits of FIG. 36B isencoded as 2 pairs of entangled physical qubits, resulting in 24 totalphysical qubits. Each of the encoded qubits is similar to the encodedqubits illustrated in FIG. 24B. The solid lines that connect physicalqubits in FIG. 36B indicate entanglement between qubits. The qubitsmarked with an “H” have a Hadamard gate applied to them.

The terminology used in the description of the various describedembodiments herein is for the purpose of describing particularembodiments only and is not intended to be limiting. As used in thedescription of the various described embodiments and the appendedclaims, the singular forms “a”, “an” and “the” are intended to includethe plural forms as well, unless the context clearly indicatesotherwise. It will also be understood that the term “and/or” as usedherein refers to and encompasses any and all possible combinations ofone or more of the associated listed items. It will be furtherunderstood that the terms “includes,” “including,” “comprises,” and/or“comprising,” when used in this specification, specify the presence ofstated features, integers, steps, operations, elements, and/orcomponents, but do not preclude the presence or addition of one or moreother features, integers, steps, operations, elements, components,and/or groups thereof.

It will also be understood that, although the terms first, second, etc.,are, in some instances, used herein to describe various elements, theseelements should not be limited by these terms. These terms are only usedto distinguish one element from another. For example, a first switchcould be termed a second switch, and, similarly, a second switch couldbe termed a first switch, without departing from the scope of thevarious described embodiments. The first switch and the second switchare both switches, but they are not the same switch unless explicitlystated as such.

As used herein, the term “if” is, optionally, construed to mean “when”or “upon” or “in response to determining” or “in response to detecting”or “in accordance with a determination that,” depending on the context.

The foregoing description, for purpose of explanation, has beendescribed with reference to specific embodiments. However, theillustrative discussions above are not intended to be exhaustive or tolimit the scope of the claims to the precise forms disclosed. Manymodifications and variations are possible in view of the aboveteachings. The embodiments were chosen in order to best explain theprinciples underlying the claims and their practical applications, tothereby enable others skilled in the art to best use the embodimentswith various modifications as are suited to the particular usescontemplated.

What is claimed is:
 1. A method for performing quantum computing, themethod comprising: performing a first fusion measurement on a firstphysical qubit of a first encoded qubit and a second physical qubit of asecond encoded qubit; obtaining a first classical measurement result ofthe first fusion measurement; selecting a basis for performing a secondfusion measurement based at least in part on the first classicalmeasurement result, wherein the selected basis determines a failurebasis for the second fusion measurement based at least in part onwhether the first classical measurement result comprises a success,failure or loss outcome; performing the second fusion measurement on athird physical qubit of the first encoded qubit and a fourth physicalqubit of the second encoded qubit according to the selected basis;obtaining a second classical measurement result of the second fusionmeasurement; determining an encoded fusion measurement result of afusion of the first encoded qubit and the second encoded qubit based atleast in part on the first and second classical measurement results; andstoring the encoded fusion measurement result in a non-transitory memorymedium, wherein selecting the basis for performing the second fusionmeasurement based at least in part on the first classical measurementresult comprises: determining that, when the second fusion measurementfails in a first basis, the encoded qubit measurement result may stillbe determined based at least in part on the first and the secondclassical measurement results; and selecting the first basis as theselected basis for performing the second fusion measurement based atleast in part on determining that, when the second fusion measurementfails in the first basis, the encoded qubit measurement result may stillbe determined based at least in part on the first and second classicalmeasurement results.
 2. The method of claim 1, wherein selecting thebasis for performing the second fusion measurement comprises:determining, based on stabilizers of the first and second encodedqubits, a plurality of effective measurement results that correspond tothe encoded fusion measurement result; determining that a firsteffective measurement result of the plurality of effective measurementresults comprises the first classical measurement result and the secondclassical measurement result when the second fusion measurement fails inthe selected basis.
 3. The method of claim 1, the method furthercomprising: determining, based at least in part on the first classicalmeasurement result, whether to employ boosting for the second fusionmeasurement.
 4. The method of claim 3, wherein determining whether toemploy boosting comprises: determining whether the encoded fusionmeasurement result is determinable when the second fusion measurementfails in the selected basis, wherein it is determined not to employboosting when the encoded fusion measurement result is determinable whenthe second fusion measurement fails in the selected basis.
 5. The methodof claim 3, wherein determining whether to employ boosting comprises:determining a photon loss rate of the second fusion measurement;determining loss probabilities for performing the second fusionmeasurement with and without boosting based on the photon loss rate;determining success probabilities for performing the second fusionmeasurement with and without boosting based on the loss probabilitiesand based on failure probabilities for performing the second fusionmeasurement with and without boosting; and determining whether to employboosting based on a comparison of the success probabilities forperforming the second fusion measurement with and without boosting. 6.The method of claim 1, wherein selecting the basis for performing thesecond fusion measurement comprises determining whether to apply aHadamard gate at a fusion site while performing the second fusionmeasurement.
 7. The method of claim 1, the method further comprising:computing an output of a quantum computational algorithm based at leastin part on the encoded fusion measurement result; and storing the outputin the non-transitory memory medium.
 8. A system, comprising: anon-transitory computer-readable memory medium; a first encoded qubitcomprising a first plurality of physical qubits; a second encoded qubitcomprising a second plurality of physical qubits; a fusion controller;and a plurality of fusion sites coupled to the fusion controller,wherein the system is configured to: perform a first fusion measurementat a first fusion site of the plurality of fusion sites on a firstphysical qubit of the first plurality of physical qubits and a secondphysical qubit of the second plurality of physical qubits; obtain afirst classical measurement result of the first fusion measurement;select a basis for performing a second fusion measurement based at leastin part on the first classical measurement result, wherein the selectedbasis determines a failure basis for the second fusion measurement basedat least in part on whether the first classical measurement resultcomprises a success, failure or loss outcome; perform the second fusionmeasurement at a second fusion site of the plurality of fusion sites ona third physical qubit of the first plurality of physical qubits and afourth physical qubit of the second plurality of physical qubitsaccording to the selected basis; obtain a second classical measurementresult of the second fusion measurement; determine an encoded fusionmeasurement result of a fusion of the first encoded qubit and the secondencoded qubit based at least in part on the first and second classicalmeasurement results; and store the encoded fusion measurement result ina non-transitory memory medium, wherein, in selecting the basis forperforming the second fusion measurement based at least in part on thefirst classical measurement result, the system is further configured to:determine that, when the second fusion measurement fails in a firstbasis, the encoded qubit measurement result may still be determinedbased at least in part on the first and the second classical measurementresults; and select the first basis as the selected basis for performingthe second fusion measurement based at least in part on determiningthat, when the second fusion measurement fails in the first basis, theencoded qubit measurement result may still be determined based at leastin part on the first and second classical measurement results.
 9. Thesystem of claim 8, wherein the system is further configured to: selectone or more bases for performing one or more subsequent fusionmeasurements on physical qubits from the first and second plurality ofphysical qubits, wherein the one or more bases are selected based atleast in part on classical measurement results of previous fusionmeasurements; perform the one or more subsequent fusion measurements;and obtain one or more subsequent classical measurement results, whereindetermining the encoded fusion measurement result is performed furtherbased at least in part on the one or more subsequent classicalmeasurement results.
 10. The system of claim 8, wherein, in selectingthe basis for performing the second fusion measurement based at least inpart on the first classical measurement result, the system is furtherconfigured to: determine, based on stabilizers of the first and secondencoded qubits, a plurality of effective measurement results thatcorrespond to the encoded fusion measurement result; determine that afirst effective measurement result of the plurality of effectivemeasurement results comprises the first classical measurement result andthe second classical measurement result when the second fusionmeasurement fails in the selected basis.
 11. The system of claim 8,wherein the system is further configured to: determine whether theencoded fusion measurement result is determinable when the second fusionmeasurement fails in the selected basis; employ boosting for the secondfusion measurement when the encoded fusion measurement result is notdeterminable when the second fusion measurement fails in the selectedbasis; and refrain from employing boosting for the second fusionmeasurement when the encoded fusion measurement result is determinablewhen the second fusion measurement fails in the selected basis.
 12. Thesystem of claim 8, wherein selecting the basis for performing the secondfusion measurement comprises determining whether to apply a Hadamardgate at the second fusion site while performing the second fusionmeasurement.
 13. The system of claim 8, wherein the system is furtherconfigured to: compute an output of a quantum computational algorithmbased at least in part on the encoded fusion measurement result; andstore the output in the non-transitory computer-readable memory medium.14. The system of claim 8, wherein the system is further configured to:determine, based at least in part on the first classical measurementresult, whether to employ boosting for the second fusion measurement,wherein in determining whether to employ boosting, the system isconfigured to: determine a photon loss rate of the second fusionmeasurement; determine loss probabilities for performing the secondfusion measurement with and without boosting based on the photon lossrate; determine success probabilities for performing the second fusionmeasurement with and without boosting based on the loss probabilitiesand based on failure probabilities for performing the second fusionmeasurement with and without boosting; and determine whether to employboosting based on a comparison of the success probabilities forperforming the second fusion measurement with and without boosting. 15.A non-transitory computer-readable memory medium storing programinstructions which, when executed by a processor, cause a fusioncontroller to: perform a first fusion measurement on a first physicalqubit of a first encoded qubit and a second physical qubit of a secondencoded qubit; obtain a first classical measurement result of the firstfusion measurement; select a first basis for performing a second fusionmeasurement based at least in part on the first classical measurementresult, wherein the selected basis determines a failure basis for thesecond fusion measurement based at least in part on whether the firstclassical measurement result comprises a success, failure or lossoutcome; perform the second fusion measurement on a third physical qubitof the second encoded qubit and a fourth physical qubit of a thirdencoded qubit according to the first basis; obtain a second classicalmeasurement result of the second fusion measurement; select a secondbasis for performing a third fusion measurement based at least in parton the first and second classical measurement results; perform the thirdfusion measurement on a fifth physical qubit of the third encoded qubitand a sixth physical qubit of the first encoded qubit according to thesecond basis; obtain a third classical measurement result of the thirdfusion measurement; determine an encoded fusion measurement result of afusion of the first, second and third encoded qubits based at least inpart on the first, second and third classical measurement results; andstore the encoded fusion measurement result in the non-transitorycomputer-readable memory medium, wherein, in selecting the first basisfor performing the second fusion measurement based at least in part onthe first classical measurement result, the program instructions arefurther executable to cause the fusion controller to: determine that,when the second fusion measurement fails in the first basis, the encodedqubit measurement result may still be determined based at least in parton the first and the second classical measurement results.
 16. Thenon-transitory computer-readable memory medium of claim 15, wherein, inselecting the second basis for performing the third fusion measurementbased at least in part on the first and second classical measurementresults, the program instructions are further executable to cause thefusion controller to: determine that, when the third fusion measurementfails in the second basis, the encoded qubit measurement result maystill be determined based at least in part on the first, second andthird classical measurement results.
 17. The non-transitorycomputer-readable memory medium of claim 15, wherein the programinstructions are further executable to cause the fusion controller to:determine whether the encoded fusion measurement result is determinablewhen the second fusion measurement fails in the first basis; employboosting for the second fusion measurement when the encoded fusionmeasurement result is not determinable when the second fusionmeasurement fails in the first basis; and refrain from employingboosting for the second fusion measurement when the encoded fusionmeasurement result is determinable when the second fusion measurementfails in the first basis.
 18. The non-transitory computer-readablememory medium of claim 15, wherein selecting the first basis forperforming the second fusion measurement comprises determining whetherto apply a first Hadamard gate while performing the second fusionmeasurement, and wherein selecting the second basis for performing thethird fusion measurement comprises determining whether to apply a secondHadamard gate while performing the third fusion measurement.
 19. Thenon-transitory computer-readable memory medium of claim 15, wherein theprogram instructions are further executable to cause the fusioncontroller to: compute an output of a quantum computational algorithmbased at least in part on the encoded fusion measurement result; andstore the output in the non-transitory computer-readable memory medium.20. The non-transitory computer-readable memory medium of claim 15,wherein the program instructions are further executable to cause thefusion controller to: determine, based at least in part on the firstclassical measurement result, whether to employ boosting for the secondfusion measurement, wherein in determining whether to employ boosting,the program instructions are further executable to cause the fusioncontroller to: determine a photon loss rate of the second fusionmeasurement; determine loss probabilities for performing the secondfusion measurement with and without boosting based on the photon lossrate; determine success probabilities for performing the second fusionmeasurement with and without boosting based on the loss probabilitiesand based on failure probabilities for performing the second fusionmeasurement with and without boosting; and determine whether to employboosting based on a comparison of the success probabilities forperforming the second fusion measurement with and without boosting.